{"id":90196,"date":"2022-03-31T15:25:29","date_gmt":"2022-03-31T09:55:29","guid":{"rendered":"https:\/\/bigideasmathanswers.com\/?p=90196"},"modified":"2022-04-07T14:31:14","modified_gmt":"2022-04-07T09:01:14","slug":"envision-math-common-core-grade-8-answer-key-topic-6","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/envision-math-common-core-grade-8-answer-key-topic-6\/","title":{"rendered":"enVision Math Common Core Grade 8 Answer Key Topic 6 Congruence And Similarity"},"content":{"rendered":"

Practice with the help of enVision Math Common Core Grade 8 Answer Key<\/a> Topic 6 Congruence and Similarity<\/strong> regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 8th Grade Answers Key Topic 6 Congruence And Similarity<\/h2>\n

Topic Essential Question<\/strong>
\nHow can you show that two figures are either congruent or similar to one another?
\nAnswer:
\nWhen two line segments have the same length, we can say that they are congruent. When two figures have the same shape and size, we can say that the two figures are congruent. These two triangles are congruent. We can also say that their side lengths are the same\u00a0and that their angle measures are the\u00a0same<\/p>\n

3-ACT MATH<\/strong><\/p>\n

\"Envision<\/p>\n

Tricks of the Trade
\nAll kinds of objects in nature have symmetry: beehives, pine cones, butterflies, and snowflakes, to name a few. If you look closely, you will start to see patterns left and right. Think about this during the 3-Act Mathematical Modeling lesson.
\n\"Envision<\/p>\n

Topic 6 \u00ebnVision STEM Project<\/strong><\/p>\n

Did You Know?
\nTrees provide wood for cooking and heating for half of the world’s population.
\n\"Envision
\nAs trees grow, carbon dioxide is removed from the atmosphere for photosynthesis. Forests are called “carbon sinks” because one acre of forest absorbs six tons of carbon dioxide and puts out four tons of oxygen.
\n\"Envision
\n\"Envision<\/p>\n

Trees provide lumber for buildings, tools, and furniture. Other products include rubber, sponges, cork, paper, chocolate, nuts, and fruit.
\nAbout 30% of the land is covered by forests.
\n\"Envision
\nForests are now being managed to preserve wildlife and old-growth forests, protect biodiversity, safeguard watersheds, and develop recreation, as well as extract timber.<\/p>\n

Forests also need to be managed to prevent raging wildfires, invasive species, overgrazing, and disease.
\n\"Envision<\/p>\n

Your Task: Forest Health
\nThe proper management of forests is a growing science. You and your classmates will learn about forest health indicators and use what you know about similar triangles and ratios to gather and interpret data in order to assess the health of a forest.
\n\"Envision<\/p>\n

Topic 6 GET READY!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary
\nChoose the best term from the box to complete each definition.
\nadjacent angles
\ncomplementary angles
\nsupplementary angles
\nvertical angles<\/p>\n

Question 1.
\n_________ have a sum of 90\u00b0.
\nAnswer:
\nWe know that,
\nThe “Complementary angles” have a sum of 90\u00b0
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Complementary angles<\/p>\n

Question 2.
\n_________ share the same ray.
\nAnswer:
\nWe know that,
\nThe “Adjacent Angles” share the same ray
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Adjacent angles<\/p>\n

Question 3.
\n_________ are pairs of opposite angles made by intersecting lines.
\nAnswer:
\nWe know that,
\nThe “Vertical Angles” are pairs of opposite angles made by intersecting lines
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Vertical angles<\/p>\n

Question 4.
\n__________ have a sum of 180\u00b0
\nAnswer:
\nWe know that,
\nThe “Supplementary Angles” have a sum of 180\u00b0
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Supplementary angles<\/p>\n

Multiplying Real Numbers<\/strong><\/p>\n

Simplify the expression.
\nQuestion 5.
\n5 \u00d7 2 = ________
\nAnswer:
\nThe given expression is:
\n5 \u00d7 2
\nHence, from the above,
\nWe can conclude that
\n5 \u00d7 2 = 10<\/p>\n

Question 6.
\n6 \u00d7 \\(\\frac{1}{2}\\) = ________
\nAnswer:
\nThe given expression is:
\n6 \u00d7 \\(\\frac{1}{2}\\)
\nSo,
\n6 \u00d7 \\(\\frac{1}{2}\\)
\n= \\(\\frac{6}{2}\\)
\n= 3
\nHence, from the above,
\nWe can conclude that
\n6 \u00d7 \\(\\frac{1}{2}\\) = 3<\/p>\n

Question 7.
\n12 \u00d7 \\(\\frac{1}{3}\\) = ________
\nAnswer:
\nThe given expression is:
\n12 \u00d7 \\(\\frac{1}{3}\\)
\nSo,
\n12 \u00d7 \\(\\frac{1}{3}\\)
\n= \\(\\frac{12}{3}\\)
\n= 4
\nHence, from the above,
\nWe can conclude that
\n12 \u00d7 \\(\\frac{1}{3}\\) = 4<\/p>\n

Identifying Points on a Coordinate Plane<\/strong><\/p>\n

Name the location of the point.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nWe can observe that
\n\"\"<\/p>\n

Question 8.
\npoint W
\nAnswer:
\nFrom the given coordinate plane,
\nWe can observe that point W lies where both the x-axis and y-axis are negative
\nHence, from the above,
\nWe can conclude that point W lies in the third quadrant<\/p>\n

Question 9.
\npoint X
\nAnswer:
\nFrom the given coordinate plane,
\nWe can observe that point X lies where the x-axis is negative and the y-axis is positive
\nHence, from the above,
\nWe can conclude that point X lies in the second quadrant<\/p>\n

Question 10.
\npoint Y
\nAnswer:
\nFrom the given coordinate plane,
\nWe can observe that point Y lies where both the x-axis and y-axis are positive
\nHence, from the above,
\nWe can conclude that point Y lies in the first quadrant<\/p>\n

Question 11.
\npoint Z
\nAnswer:
\nFrom the given coordinate plane,
\nWe can observe that point Z lies where the x-axis is positive and the y-axis is negative
\nHence, from the above,
\nWe can conclude that point Z lies in the fourth quadrant<\/p>\n

Supplementary Angles<\/strong><\/p>\n

The angles are supplementary. Find the missing angle measure.
\nQuestion 12.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nWe know that,
\nThe sum of the supplementary angles is: 180\u00b0
\nNow,
\nLet the missing angle measure be: x\u00b0
\nSo,
\nx\u00b0 + 130\u00b0 = 180\u00b0
\nx\u00b0 = 180\u00b0 – 130\u00b0
\nx\u00b0 = 50\u00b0
\nHence, from the above,
\nWe can conclude that the missing angle measure is: 50\u00b0<\/p>\n

Question 13.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nWe know that,
\nThe sum of the supplementary angles is: 180\u00b0
\nNow,
\nLet the missing angle measure be: x\u00b0
\nSo,
\nx\u00b0 + 139\u00b0 = 180\u00b0
\nx\u00b0 = 180\u00b0 – 139\u00b0
\nx\u00b0 = 41\u00b0
\nHence, from the above,
\nWe can conclude that the missing angle measure is: 41\u00b0<\/p>\n

Language Development<\/strong>
\nComplete the graphic organizer with an illustration for each transformation. Write either congruent or similar to make the given statement true.
\n\"Envision
\n\"Envision
\nAnswer:
\n\"\"<\/p>\n

Topic 6 PICK A PROJECT<\/h3>\n

PROJECT 6A
\nHow might an artist use mathematics?
\nPROJECT: WRITE A BIOGRAPHY
\n\"Envision<\/p>\n

PROJECT 6B
\nWhat geometric shapes do you see around you?
\nPROJECT: RECORD A VIDEO ABOUT SIMILAR FIGURES
\n\"Envision<\/p>\n

PROJECT 6C
\nWhat different types of bridges have you crossed?
\nPROJECT: BUILD A MODEL OF A TRUSS BRIDGE
\n\"Envision<\/p>\n

PROJECT 6D
\nWhat shapes tessellate?
\nPROJECT: DESIGN A TESSELLATION
\n\"Envision<\/p>\n

Lesson 6.1 Analyze Translations<\/h3>\n

Solve & Discuss It!<\/strong>
\nAshanti draws a trapezoid on the coordinate plane and labels it in Figure 1. Then she draws Figure 2. How can she determine whether the figures have the same side lengths and the same angle measures?
\n\"Envision
\nI can… translate two-dimensional figures.
\nAnswer:
\nIt is given that
\nAshanti draws a trapezoid on the coordinate plane and labels it in Figure 1. Then she draws Figure 2.
\nThe given figure is:
\n\"Envision
\nFrom the figure,
\nWe can observe that the first figure was translated or moved to the right by some units and became a second figure
\nWe know that,
\nA “Translation” is a transformation that moves every point in a figure the same distance in the same direction
\nWe know that,
\nThe pre-image and image in the “Translation” are similar
\nWe know that,
\nTwo figures are said to be\u00a0similar if they are the same shape. In more\u00a0mathematical\u00a0language, two figures are\u00a0similar\u00a0if their corresponding angles are congruent, and the ratios of the lengths of their corresponding sides are equal.
\nHence, from the above,
\nWe can conclude that by using the “Translation” property, she can determine whether the figures have the same side lengths and the same angle measures<\/p>\n

Focus on math practices
\nBe Precise How do you know that the method you described shows whether the side lengths and angle measures are equal? Explain.
\nAnswer:
\nWe know that,
\nA “Translation” is a transformation that moves every point in a figure the same distance in the same direction
\nWe know that,
\nThe pre-image and image in the “Translation” are similar
\nWe know that,
\nTwo figures are said to be\u00a0similar if they are the same shape. In more\u00a0mathematical\u00a0language, two figures are\u00a0similar\u00a0if their corresponding angles are congruent, and the ratios of the lengths of their corresponding sides are equal.<\/p>\n

Essential Question<\/strong>
\nHow does a translation affect the properties of a two-dimensional figure?
\nAnswer:
\nWhen you\u00a0translate\u00a0something in geometry, you’re simply moving it around. You don’t distort it in any way. If you\u00a0translate\u00a0a segment, it remains a segment, and its length doesn’t change. Similarly, if you\u00a0translate\u00a0an angle, the measure of the angle doesn’t change.<\/p>\n

Try It!<\/strong><\/p>\n

The clients also want the small table below the window moved 5 feet to the right. Where should the architect place the small table? Draw the new location of the table on the plan.
\nAnswer:
\nIt is given that
\nThe clients also want the small table below the window moved 5 feet to the right
\nNow,
\nThe original plan is:
\n\"\"
\nFrom the above plan,
\nWe can observe that the window is at most right
\nSo,
\nWhen we move the table,
\nThe new location of the table will be below the window
\nHence,
\nThe location of the table on the new plan is:
\n\"\"<\/p>\n

Convince Me!
\nAn equilateral triangle with side lengths 5 inches is translated 3 units down and 2 units right. Describe the shape and dimensions of the translated figure.
\nAnswer:
\nIt is given that
\nAn equilateral triangle with side lengths 5 inches is translated 3 units down and 2 units right.
\nWe know that,
\nIn an equilateral triangle, all the side lengths are equal
\nNow,
\nLet the dimensions of an equilateral triangle be (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>), and (x3<\/sub>, y3<\/sub>)
\nSo,
\nThe dimensions of the translated figure are: (x1<\/sub> + 2, y1<\/sub> – 3), (x2<\/sub> + 2, y2<\/sub> – 3), and (x3<\/sub> + 2, y3<\/sub> – 3)
\nNow,
\nLet us suppose the dimensions of an equilateral to be:
\n(1, 1), (1, 6), and (5, 3)
\nSo,
\nThe dimensions of the translated equilateral triangle are:
\n(3, -2), (3, 3), and (7, 0)
\nWe know that,
\nWhen you\u00a0translate\u00a0something in geometry, you’re simply moving it around. You don’t distort it in any way
\nSo,
\nThe shape of the translated figure remains the same as the original figure
\nHence,
\nThe representation and shape of the original and translated equilateral triangles are:
\n\"\"<\/p>\n

Try It!<\/strong><\/p>\n

Triangle ABC is translated 5 units right and 1 unit down. Graph and label the image A’ B’C’. If m\u2220A = 30\u00b0 in \u0394ABC, then what is m\u2220A in \u0394A’B’C’?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nFrom the given figure,
\nThe dimensions of \u0394ABC are:
\nA (-2, 3), B (-3, -4), and C (-5, -1)
\nNow,
\nLet the dimensions of an equilateral triangle be (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>), and (x3<\/sub>, y3<\/sub>)
\nSo,
\nThe dimensions of the translated figure are: (x1<\/sub> + 5, y1<\/sub> – 1), (x2<\/sub> + 5, y2<\/sub> – 1), and (x3<\/sub> + 5, y3<\/sub> – 1)
\nSo,
\nThe dimensions of the translated triangle (A’B’C’) are:
\nA (3, 2), B (2, -5), and C (0, -2)
\nWe know that,
\nA translated figure has the same shape as the original shape i.e, the lengths and the angles of the translated figure are also the same as the original figure
\nSo,
\nm\u2220A in Triangle ABC = m\u2220A in Triangle A’B’C’ = 30\u00b0
\nHence,
\nThe representation of the original and translated triangles are:
\n\"\"<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

A translation, or slide, is a transformation that moves Preimage to every point of a figure the same distance and the same direction.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How does a translation affect the properties of a two-dimensional figure?
\nAnswer:
\nWhen you\u00a0translate\u00a0something in geometry, you’re simply moving it around. You don’t distort it in any way. If you\u00a0translate\u00a0a segment, it remains a segment, and its length doesn’t change. Similarly, if you\u00a0translate\u00a0an angle, the measure of the angle doesn’t change.<\/p>\n

Question 2.
\nConstruct Arguments<\/strong> Triangle L’M’N’ is the image of triangle LMN after a translation. How are the side lengths and angle measures of the triangles related? Explain.
\nAnswer:
\nWe know that,
\nWhen you\u00a0translate\u00a0something in geometry, you’re simply moving it around. You don’t distort it in any way. If you\u00a0translate\u00a0a segment, it remains a segment, and its length doesn’t change. Similarly, if you\u00a0translate\u00a0an angle, the measure of the angle doesn’t change.
\nHence, from the above,
\nWe can conclude that
\nThe side lengths and the angle measures of Triangle L’M’N’ is the same as The side lengths and the angle measures of Triangle LMN<\/p>\n

Question 3.
\nGeneralize<\/strong> Sanjay determined that one vertex of a figure was mapped to its image by translating the point 2 units left and 7 units down. What is the rule that maps the other vertices of the figure to their images?
\nAnswer:
\nIt is given that
\nSanjay determined that one vertex of a figure was mapped to its image by translating the point 2 units left and 7 units down
\nNow,
\nNow,
\nLet the dimensions of any vertex of the given be (x, y)
\nSo,
\nThe dimensions of any vertex of the translated figure are: (x – 2, y + 7)
\nHence, from the above,
\nWe can conclude that
\nThe rule that maps the other vertices of the figure to their images is: (x – 2, y – 7)<\/p>\n

Do You Know How?<\/strong>
\nIn 4-6, use the coordinate plane.
\n\"Envision<\/p>\n

Question 4.
\nWhich figure is a translation of Figure A? Explain
\nAnswer:
\nWe know that,
\nWhen you\u00a0translate\u00a0something in geometry, you’re simply moving it around. You don’t distort it in any way. If you\u00a0translate\u00a0a segment, it remains a segment, and its length doesn’t change. Similarly, if you\u00a0translate\u00a0an angle, the measure of the angle doesn’t change.
\nSo,
\nWhen we observe the coordinate plane,
\nWe can see that A and C have the same shape i.e., have the same length and the same angle
\nHence, from the above,
\nWe can conclude that
\nFigure C is a translation of Figure A<\/p>\n

Question 5.
\nGraph the translation of Figure A 3 units right and 4 units up.
\nAnswer:
\nFrom the given coordinate plane,
\nThe dimensions of Figure A are:
\n(-2, -1), (-4, -1), (-4, -2), (-3, -2), (-3, -3), (-4, -3), (-4, -4), and (-2, -4)
\nSo,
\nThe rule that maps the vertices of the given figure to their images is: (x + 3, y + 4)
\nSo,
\nThe vertices of the image are:
\n(1, 3), (-1, 3), (-1, 2), (0, 2), (0, 1), (-1, 1), (-1, 0), and (1, 0)
\nHence,
\nThe representation of Figure A and its translated figure is:
\n\"\"<\/p>\n

Question 6.
\nDescribe the translation needed to move Figure B to the same position as the image from Item 5.
\nAnswer:
\nFrom the given coordinate plane,
\nThe dimensions of figure B are:
\n(-2, 4), (-4, 4), (-4, 3), (-3, 3), (-3, 2), (-4, 2), (-4, 1), and (-2, 1)
\nHence,
\nThe translation needed to move figure B to the same position as the image from Item 5 is:
\nThe X-axis: Translate 3 units right
\nThe Y-axis: Translate 1 unit down<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nGraph G’R’A’M’, the image of GRAM after a translation 11 units right and 2 units up.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe dimensions of GRAM are:
\nG (-8, 2), R (-8, 6), A (-4, 6), and M (-5, 2)
\nSo,
\nAfter a translation of 11 units right and 2 units up,
\nThe dimensions of G’R’A’M’ are:
\nG’ (-8 + 11, 2 + 2), R’ (-8 + 11, 6 + 2), A’ (-4 + 11, 6 + 2), and M’ (-5 + 11, 2 + 2)
\nSo,
\nG’ (3, 4), R’ (3, 8), A’ (7, 8), and M’ (6, 4)
\nHence,
\nThe representation of GRAM and its image G’R’A’M’ is:
\n\"\"<\/p>\n

Question 8.
\n\u2206A’ B’ C’ is a translation of \u2206ABC. Describe the translation.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the coordinate plane,
\nThe dimensions of \u0394ABC are:
\nA (1, -5), B (-2, -2), and C (0, 0)
\nNow,
\nThe dimensions of \u0394A’B’C’ are:
\nA’ (0, -1), B’ (-3, 2), and C’ (-1, 4)
\nSo,
\nBy observing the dimensions of \u0394ABC and \u0394A’B’C’,
\nThe translation between \u0394ABC and \u0394A’B’C is:
\nThe x-axis: Translate 1 unit left
\nThe y-axis: Translate 4 units up<\/p>\n

Question 9.
\nWhich triangle is the image of \u2206DEF after a translation? Describe the translation.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nWe know that,
\nWhen you\u00a0translate\u00a0something in geometry, you’re simply moving it around. You don’t distort it in any way. If you\u00a0translate\u00a0a segment, it remains a segment, and its length doesn’t change. Similarly, if you\u00a0translate\u00a0an angle, the measure of the angle doesn’t change.
\nSo,
\nFrom the given figure,
\nWe can observe that
\n\u0394MNO has the same shape as \u0394DEF
\nSo,
\nThe image of \u0394DEF is: \u0394MNO
\nNow,
\nThe dimensions of \u0394DEF are:
\nD (-8, -10), F (-6, -10), and E (-8, -4)
\nNow,
\nThe dimensions of \u0394MNO are:
\nM (2, -10), O (4, -10), and N (2, -4)
\nSo,
\nBy observing the dimensions of \u0394DEF and \u0394MNo,
\nThe translation between \u0394DEF and \u0394MNO is:
\nThe x-axis: Translate 10 units right
\nThe y-axis: No Translation required<\/p>\n

Question 10.
\nThe vertices of figure QRST are translated 3 units left and 11 units down to form figure Q’R’S’T’. Explain the similarities and differences between the two figures.
\nAnswer:
\nIt is given that
\nThe vertices of figure QRST are translated 3 units left and 11 units down to form figure Q’R’S’T’.
\nHence,
\nThe similarities between figure QRST and figure Q’R’S’T’ are:
\na. The two figures have the same side length
\nb. The two figures have the same angle measure
\nc. The two figures have the same shape
\nThe differences between figure QRST and figure Q’R’S’T’ are:
\na. Different dimensions of the vertices of QRST and Q’R’S’T’
\nb. Different positions of figure QRST and figure Q’R’S’T’<\/p>\n

Question 11.
\nGraph the image of the given triangle after a translation of 3 units right and 2 units up.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe dimensions of the given triangle are:
\n(-3, 4), (-5, -1), and (-8, 3)
\nNow,
\nAfter a translation 3 units right and 2 units up,
\nThe dimensions of the given triangle are:
\n(-3 + 3, 4 + 2), (-5 + 3, -1 + 2), and (-8 + 3, 3 + 2)
\n(0, 6), (-2, 1), and (-5, 5)
\nHence,
\nThe representation of the given triangle and its image is:
\n\"\"<\/p>\n

Question 12.
\nQuadrilateral P’Q’R’ S’ is the image of quadrilateral PQRS after a translation.
\n\"Envision
\nAnswer:
\nIt is given that
\nQuadrilateral P’Q’R’ S’ is the image of quadrilateral PQRS after a translation.
\nWe know that,
\nWhen you\u00a0translate\u00a0something in geometry, you’re simply moving it around. You don’t distort it in any way. If you\u00a0translate\u00a0a segment, it remains a segment, and its length doesn’t change. Similarly, if you\u00a0translate\u00a0an angle, the measure of the angle doesn’t change.<\/p>\n

a. If the length of side PQ is about 2.8 units, what is the length of side P’ Q’?
\nAnswer:
\nIt is given that
\nThe length of the side PQ is about 2.8 units
\nHence, from the above,
\nWe can conclude that the length of side P’Q’ is also about 2.8 units<\/p>\n

b. If m\u2220R = 75\u00b0, what is m\u2220R’?
\nAnswer:
\nIt is given that
\nm\u2220R = 75\u00b0
\nHence, from the above,
\nWe can conclude that
\nm\u2220R’ = 75\u00b0<\/p>\n

Question 13.
\nHigher-Order Thinking<\/strong> A farmer has a plot of land shaped like the figure in the graph. There is another identical plot of land 120 yards east and 100 yards north of the original plot.
\n\"Envision
\na. Draw the image after the given translation.
\nAnswer:
\nIt is given that
\nA farmer has a plot of land shaped like the figure in the graph. There is another identical plot of land 120 yards east and 100 yards north of the original plot.
\nNow,
\nThe given plot of land is:
\n\"Envision
\nNow,
\nFrom the given plot of land,
\nThe dimensions of land are:
\n(0, 0), (0, 300), (300, 0), and (300, 300)
\nNow,
\nAfter the translation of 120 units right and 100 units up,
\nThe dimensions of the identical plot are:
\n(0 + 120, 0 + 100), (0 + 120, 300 + 100), (300 + 120, 0 + 100), and (300 + 120, 300 + 100)
\n(120, 100), (120, 400), (420, 100), and (420, 400)
\nHence,
\nThe representation of the plot and its image after the translation is:
\n\"\"<\/p>\n

b. Find the combined area of the 2 plots in square yards.
\nAnswer:
\nFrom part (a),
\nWe can observe that the shape of the plot of land and its identical is like a square
\nWe know that,
\nThe square has the equal side lengths
\nNow,
\nFrom part (a),
\nWe can observe that the side length of the plot of land and its identical is: 300 yards
\nNow,
\nWe know that,
\nThe area of square = Side\u00b2
\nSo,
\nThe area of the plot of land = 300\u00b2
\n= 90,000 yard\u00b2
\nThe area of the identical plot of land = 300\u00b2
\n= 90,000 yard\u00b2
\nSo,
\nThe combined area of the 2 plots = 90,000 + 90,000
\n= 1,80,000 yard\u00b2
\nHence, from the above,
\nWe can conclude that the combined area of the 2 plots is: 1,80,000 yard\u00b2<\/p>\n

Assessment Practice<\/strong>
\nQuestion 14.
\nWhat is true about the preimage of a figure and its image created by a translation? Select all that apply.
\n\u2610 Each point in the image moves the same distance and direction from the preimage.
\n\u2610 Each point in the image has the same x-coordinate as the corresponding point in the preimage.
\n\u2610 Each point in the image has the same y-coordinate as the corresponding point in the preimage.
\n\u2610 The preimage and the image are the same size.
\n\u2610 The preimage and the image are the same shape.
\nAnswer:
\nWe know that,
\nWhen you\u00a0translate\u00a0something in geometry, you’re simply moving it around. You don’t distort it in any way. If you\u00a0translate\u00a0a segment, it remains a segment, and its length doesn’t change. Similarly, if you\u00a0translate\u00a0an angle, the measure of the angle doesn’t change.
\nHence,
\nThe statements that are true about the preimage of a figure and its image created by a translation are:
\n\"\"<\/p>\n

Question 15.
\nThe vertices of parallelogram QUAD are Q(-7, -7), U(-6, -4), A(-2,-4), and D(-3, -7).
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe dimensions of the parallelogram QUAD are:
\nQ (-7, -7), U (-6, -4), A (-2, -4), and D (-3, -7)
\nNow,
\nAfter the translation of 11 units right and 9 units up,
\nThe dimensions of the parallelogram QUAD are:
\nQ’ (-7 + 11, -7 + 9), U’ (-6 + 11, -4 + 9), A’ (-2 + 11, -4 + 9), and D’ (-3 + 11, -7 + 9)
\nQ’ (4, 2), U’ (5, 5), A’ (9, 5), and D’ (8, 2)<\/p>\n

PART A
\nGraph and label the image of QUAD after a translation 11 units right and 9 units up.
\nAnswer:
\nThe representation of the parallelogram QUAd and its image is:
\n\"\"<\/p>\n

PART B
\nIf m\u2220U = 110\u00b0, what is m\u2220 U’?
\nAnswer:
\nWe know that,
\nWhen you\u00a0translate\u00a0something in geometry, you’re simply moving it around. You don’t distort it in any way. If you\u00a0translate\u00a0a segment, it remains a segment, and its length doesn’t change. Similarly, if you\u00a0translate\u00a0an angle, the measure of the angle doesn’t change.
\nHence, from the above,
\nWe can conclude that
\nm\u2220U’ = 110\u00b0<\/p>\n

PART C
\nIf the length of side UA is 4 units, what is the length of side U’ A’?
\nAnswer:
\nWe know that,
\nWhen you\u00a0translate\u00a0something in geometry, you’re simply moving it around. You don’t distort it in any way. If you\u00a0translate\u00a0a segment, it remains a segment, and its length doesn’t change. Similarly, if you\u00a0translate\u00a0an angle, the measure of the angle doesn’t change.
\nHence, from the above,
\nWe can conclude that
\nThe length of the side U’ A’ is: 4 units<\/p>\n

Lesson 6.2 Analyze Reflections<\/h3>\n

Solve & Discuss It!<\/strong>
\nDale draws a triangle on grid paper and labels it in Figure 1. Then using his pencil as a guide, he draws another triangle directly on the opposite side of the pencil so that the vertical side is now one square to the right of the pencil instead of one square to the left of the pencil. He labels this triangle in Figure 2. How are the figures the same? How are they different?
\n\"Envision
\nI can… reflect two-dimensional figures.
\nAnswer:
\nDale draws a triangle on grid paper and labels it in Figure 1. Then using his pencil as a guide, he draws another triangle directly on the opposite side of the pencil so that the vertical side is now one square to the right of the pencil instead of one square to the left of the pencil. He labels this triangle in Figure 2.
\nHence,
\nThe representation of Figure 1 and Figure 2 is:
\n\"\"
\nSo,
\nFrom the above representation,
\nThe similarities between Figure 1 and Figure 2 are:
\na. Both the figures have the same side lengths
\nb. Both the figures have the same angle measures
\nc. Both the figures have the same size
\nd. Both the figures have the same distance
\nThe differences between Figure 1 and Figure 2 are:
\na. The positions of both figures are different
\nb. The orientations of both figures are different
\nc. The directions of both the figures are different<\/p>\n

Look for Relationships
\nWhat do you notice about the size, shape, and direction of the two figures?
\nAnswer:
\nThe similarities between Figure 1 and Figure 2 are:
\na. Both the figures have the same side lengths
\nb. Both the figures have the same angle measures
\nc. Both the figures have the same size
\nd. Both the figures have the same distance
\nThe differences between Figure 1 and Figure 2 are:
\na. The positions of both figures are different
\nb. The orientations of both figures are different
\nc. The directions of both the figures are different<\/p>\n

Focus on math practices
\nReasoning Dale draws a line in place of his pencil and folds the grid paper along the line. How do the triangles align when the grid paper is folded? Explain.
\nAnswer:
\nIt is given that
\nDale draws a line in place of his pencil and folds the grid paper along the line
\nNow,
\nThe representation of Figure 1 and Figure 2 are:
\n\"\"
\nNow,
\nFrom the given figure,
\nWe can observe that the two triangles will stack on each other when the grid paper is folded i.e.,
\na. The vertical side of figure 2 is on top of the vertical side of figure 1
\nb. The base of figure 2 is on top of the base of figure 1
\nc. The hypotenuse of figure 2 is on top of the hypotenuse of figure 1<\/p>\n

Essential Question<\/strong>
\nHow does a reflection affect the properties of a two-dimensional figure?
\nAnswer:
\nWhen the reflection takes place along the x-axis,
\nThe values of x will remain constant and the values of y will have a sign change
\nEx:
\nWhen (x, y) and (-x, y) reflects along the x-axis,
\nThe reflection of (x, y) will become (x, -y)
\nThe reflection of (-x, y) will become (-x, -y)
\nWhen the reflection takes place along the y-axis,
\nThe values of y will remain constant and the values of x will have a sign change
\nEx:
\nWhen (x, y) and (x, -y) reflects along the y-axis,
\nThe reflection of (x, y) will become (-x, y)
\nThe reflection of (x, -y) will become (-x, -y)<\/p>\n

Try It!<\/strong><\/p>\n

While updating the design, the architect accidentally clicked on the chair and reflected it across the centerline. Draw the new location of the chair on the plan.
\n\"\"
\nAnswer:
\nIt is given that
\nWhile updating the design, the architect accidentally clicked on the chair and reflected it across the centerline.
\nHence,
\nThe representation of the new location of the chair is:
\n\"\"<\/p>\n

Convince Me!
\nHow do the preimage and image compare after a reflection?
\nAnswer:
\nA reflection is a transformation that turns a figure into its mirror image by flipping it over a line. The line of reflection is the line that a figure is reflected over. If a point is on the line of reflection then the image is the same as the preimage. Otherwise,
\nthe image is not the same as the preimage. Images are always congruent to preimages<\/p>\n

Try It!<\/strong><\/p>\n

Quadrilateral KLMN has vertices at K(2, 6), L(3, 8), M(5, 4), and N(3, 2). It is reflected across the y-axis, resulting in quadrilateral K’L’M’N’. What are the coordinates of point N’?
\nAnswer:
\nIt is given that
\nQuadrilateral KLMN has vertices at K(2, 6), L(3, 8), M(5, 4), and N(3, 2). It is reflected across the y-axis, resulting in quadrilateral K’L’M’N’
\nNow,
\nWe know that,
\nWhen (x, y) and (x, -y) reflects along the y-axis,
\nThe reflection of (x, y) will become (-x, y)
\nThe reflection of (x, -y) will become (-x, -y)
\nSo,
\nThe reflection of N (3, 2) i.e., the coordinates of N’ is: (-3, 2)
\nHence, from the above,
\nWe can conclude that the coordinates of N’ are: (-3, 2)<\/p>\n

Try It!<\/strong><\/p>\n

Polygon ABCDE is reflected across the line x = -2. Graph and label the image A’B’C’D’E’. Is m\u2220A= M\u2220A? Explain.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of polygon ABCDE are:
\nA (-4, 4), B (-3, 3), C (-3, 2), D (-5, 1), and E (-5, 3)
\nIt is given that polygon ABCDE is reflected along x = -2 i.e., alone the x-axis
\nWe know that,
\nWhen (x, y) and (-x, y) reflects along the y-axis,
\nThe reflection of (x, y) will become (x, -y)
\nThe reflection of (-x, y) will become (-x, -y)
\nSo,
\nThe vertices of the reflection of polygon ABCDE i..e, A’B’C’D’E’ are:
\nA’ (-4, -4), B’ (-3, -3), C’ (-3, -2), D’ (-5, -1), and E’ (-5, -3)
\nWe know that,
\nIn reflection,
\nThe side lengths and the angle measures in the image and the preimage are the same
\nHence,
\nThe representation of polygon ABCDE and its reflection polygon A’B’C’D’E’ is:
\n\"\"
\nHence,
\nm \u2220A = M \u2220A<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

A reflection, or flip, is a transformation that flips a figure across a line of reflection. The preimage and image are the same distance from the line of reflection but on opposite sides. They have the same size and shape but different orientations.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How does a reflection affect the properties of a two-dimensional figure?
\nAnswer:
\nWhen the reflection takes place along the x-axis,
\nThe values of x will remain constant and the values of y will have a sign change
\nEx:
\nWhen (x, y) and (-x, y) reflects along the x-axis,
\nThe reflection of (x, y) will become (x, -y)
\nThe reflection of (-x, y) will become (-x, -y)
\nWhen the reflection takes place along the y-axis,
\nThe values of y will remain constant and the values of x will have a sign change
\nEx:
\nWhen (x, y) and (x, -y) reflects along the y-axis,
\nThe reflection of (x, y) will become (-x, y)
\nThe reflection of (x, -y) will become (-x, -y)<\/p>\n

Question 2.
\nGeneralize<\/strong> What do you notice about the corresponding coordinates of the preimage and image after a reflection across the x-axis?
\nAnswer:
\nWhen the reflection takes place along the x-axis,
\nThe values of x will remain constant and the values of y will have a sign change
\nEx:
\nWhen (x, y) and (-x, y) reflects along the x-axis,
\nThe reflection of (x, y) will become (x, -y)
\nThe reflection of (-x, y) will become (-x, -y)<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Jorge said the y-values would stay the same when you reflect a preimage across the line y = 5 since the y-values stay the same when you reflect a preimage across the y-axis. Is Jorge correct? Explain.
\nAnswer:
\nIt is given that
\nJorge said the y-values would stay the same when you reflect a preimage across the line y = 5 since the y-values stay the same when you reflect a preimage across the y-axis.
\nNow,
\nWe know that,
\nWhen the reflection takes place along the y-axis,
\nThe values of y will remain constant and the values of x will have a sign change
\nEx:
\nWhen (x, y) and (x, -y) reflects along the y-axis,
\nThe reflection of (x, y) will become (-x, y)
\nThe reflection of (x, -y) will become (-x, -y)
\nHence, from the above,
\nWe can conclude that Jorge is correct<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nIs AX’ Y’ Z’ a reflection of AXYZ across line g?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that \u0394XYZ is reflected across the line g i.e., y-axis
\nWe know that,
\nWhen the reflection takes place along the y-axis,
\nThe values of y will remain constant and the values of x will have a sign change
\nEx:
\nWhen (x, y) and (x, -y) reflects along the y-axis,
\nThe reflection of (x, y) will become (-x, y)
\nThe reflection of (x, -y) will become (-x, -y)
\nSo,
\nFrom the figure,
\nWe can observe that
\nThe negative x-coordinates of the vertices of \u0394XYZ became the positive x -coordinates for \u0394 X’Y’Z’
\nHence, from the above,
\nWe can conclude that \u0394X’Y’Z’ is the reflection of \u0394XYZ across the line g<\/p>\n

Use the coordinate grid below for 5 and 6.
\n\"Envision<\/p>\n

Question 5.
\nDescribe the reflection of figure EFGH.
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nWe can observe that
\nThe reflection of figure EFGH is: Figure E’F’G’H’
\nNow,
\nWe can observe that
\nThe reflection of the figure EFGH takes place across the x-axis
\nSo,
\nThe x-coordinates of the vertices of the reflection of the figure EFGH will be constant and only y-coordinates will have a change in value
\nHence,
\nThe figure EFGH will flip i.e., top becomes down and vice-versa to form a reflection i.e., figure E’F’G’H’<\/p>\n

Question 6.
\nDraw the image that would result from a reflection of figure E’F’G’H across the line x = -1.
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the reflection of EFGH i.e., figure E’F’G’H’,
\nWe can observe that the vertices of the figure E’F’G’H’ are:
\nE’ (-8, 7), F’ (-5, 5), G’ (-4, 6), and H’ (-2, 6)
\nNow,
\nTo form the image of the figure E’F’G’H’,
\nWe need to reflect the figure E’F’G’H’ across the y-axis
\nWhen the reflection takes place along the x-axis,
\nThe values of x will remain constant and the values of y will have a sign change
\nEx:
\nWhen (x, y) and (-x, y) reflects along the x-axis,
\nThe reflection of (x, y) will become (x, -y)
\nThe reflection of (-x, y) will become (-x, -y)
\nSo,
\nThe vertices for the image of the figure E’F’G’H’ are:
\nE’ (-8, -7), F’ (-5, -5), G’ (-4, -6), and H’ (-2, -6)
\nHence,
\nThe representation of the figure E’F’G’H’ and its image is:
\n\"\"<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nLeveled Practice Trapezoid ABCD is shown. Draw the reflection of trapezoid ABCD across the y-axis.
\n\"Envision
\nPlot the points and draw trapezoid A’ B’C’D’.
\nAnswer:
\nFrom the given trapezoid ABCD,
\nThe vertices are:
\nA (2, 8), B (6, 8), C (8, 3), and D (1, 3)
\nIt is given that
\nDraw the reflection of trapezoid ABCD across the y-axis i.e., y is constant
\nNow,
\nWe know that,
\nWhen the reflection takes place along the y-axis,
\nThe values of y will remain constant and the values of x will have a sign change
\nEx:
\nWhen (x, y) and (x, -y) reflects along the y-axis,
\nThe reflection of (x, y) will become (-x, y)
\nThe reflection of (x, -y) will become (-x, -y)
\nSo,
\nThe vertices for the reflection of trapezoid ABCD are:
\nA’ (-2, 8), B (-6, 8), C’ (-8, 3), and D (-1, 3)
\nSo,
\nThe points of the preimage and image are:
\n\"\"
\nHence,
\nThe representation of trapezoid A’B’C’D’ is:
\n\"\"<\/p>\n

Question 8.
\nReasoning is triangle A’ B’C’a reflection of triangle ABC across the line? Explain.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that the reflection of \u0394ABC takes place across the y-axis
\nNow,
\nWhen the reflection takes place along the y-axis,
\nThe values of y will remain constant and the values of x will have a sign change
\nEx:
\nWhen (x, y) and (x, -y) reflects along the y-axis,
\nThe reflection of (x, y) will become (-x, y)
\nThe reflection of (x, -y) will become (-x, -y)
\nSo,
\nFrom the vertices of \u0394ABC,
\nThe x-coordinates are negative and the y-coordinates are positive
\nSo,
\nFor the reflection of \u0394ABC along the y-axis,
\nThe x-coordinates will have to become positive and the y-coordinates will be positive as in the vertices of \u0394ABC
\nHence, from the above,
\nWe can conclude that \u0394A’B’C’ is the reflection of \u0394ABC across the given line<\/p>\n

Question 9.
\nYour friend gives you the graph of quadrilateral ABCD and its image, quadrilateral A’B’C’D’. What reflection produces this image?
\n\"Envision
\nAnswer:
\nIt is given that
\nYour friend gives you the graph of quadrilateral ABCD and its image, quadrilateral A’B’C’D’
\nNow,
\nThe quadrilateral ABCD and its image quadrilateral A’B’C’D’ is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe quadrilateral ABCD and its image will have the same y-coordinates but the x-coordinates are different
\nWe know that,
\nWhen the reflection takes place across the y-axis,
\nThe y-coordinates are the same for the image and the preimage
\nThe x-coordiantes have sign changes with the same values for the image and the preimage
\nHence, from the above,
\nWe can conclude that the reflection across the y-axis produces the image quadrilateral A’B’C’D’<\/p>\n

Question 10.
\nConstruct Arguments Your friend incorrectly says that the reflection of \u2206EFG to its image \u2206E’ F’G’ is a reflection across the y-axis.
\n\"Envision
\na. What is your friend’s mistake?
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the above figure,
\nWe can observe that
\nFor the vertices of the given figure,
\nThe x-coordinates remain the same whereas the y-coordinates are different
\nWe know that,
\nWhen the reflection takes place across the x-axis,
\nThe x-coordinates of the preimage and the image are the same
\nThe y-coordinates of the preimage and the image will have sign change with the same values
\nHence, from the above,
\nWe can conclude that the reflection of \u0394EFG takes pace across the x-axis instead of across the y-axis<\/p>\n

b. What is the correct description of the reflection?
\nAnswer:
\nFrom the given figure,
\nWe can observe that
\nFor the vertices of the given figure,
\nThe x-coordinates remain the same whereas the y-coordinates are different
\nWe know that,
\nWhen the reflection takes place across the x-axis,
\nThe x-coordinates of the preimage and the image are the same
\nThe y-coordinates of the preimage and the image will have sign change with the same values
\nHence, from the above,
\nWe can conclude that the reflection of \u0394EFG to its image \u0394E’F’G’ takes pace across the x-axis<\/p>\n

Question 11.
\nMake Sense and Persevere The vertices of \u2206ABC are A(-5, 5), B(-2,5), and C(-2, 3). If \u2206 ABC is reflected across the line y = -1, find the coordinates of the vertex C’
\nAnswer:
\nIt is given that
\nThe vertices of \u2206ABC are A(-5, 5), B(-2,5), and C(-2, 3) and \u2206 ABC is reflected across the line y = -1
\nWe know that,
\nWhen the reflection takes place across the y-axis,
\nThe y-coordinates of the preimage and the image are the same
\nThe x-coordinates of the preimage and the image will have sign change with the same values
\nSo,
\nThe reflection of the vertex C (-2, 3) is: (2, 3)
\nHence, from the above,
\nWe can conclude that the coordinates of the vertex C’ i.e., the reflection of the vertex C is: (2, 3)<\/p>\n

Question 12.
\nHigher Order Thinking<\/strong> What reflection of the parallelogram ABCD results in image A’B’C’D?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nFrom the given figure,
\nWe can observe that the parallelogram ABCD and its reflection A’B’C’D’ are parallel to the x-axis
\nNow,
\nWe can observe that,
\nFor both the parallelogram ABCD and its reflection A’B’C’D’,
\nThe x-coordinates are changing but the y-coordinates remain the same
\nNow,
\nWe know that,
\nWhen the reflection takes place across the y-axis,
\nThe y-coordinates of the preimage and the image are the same
\nThe x-coordinates of the preimage and the image will have sign change with the same values
\nHence, from the above,
\nWe can conclude that the reflection across the y-axis of the parallelogram ABCD results in image A’B’C’D<\/p>\n

Assessment Practice<\/strong>
\nQuestion 13.
\n\u2206JAR has vertices J(4, 5), A6, 4), and R(5,2). What graph shows \u2206JAR and its image after a reflection across the line y = 1?
\nPART A
\n\"Envision
\nAnswer:
\nIt is given that
\n\u2206JAR has vertices J(4, 5), A6, 4), and R(5,2)
\nNow,
\nWe know that,
\nThe reflection across the y-axis is:
\nThe y -coordinates are the same
\nThe x-coordinates will have sign changes with the same value
\nSo,
\nThe reflection of the vertices of \u0394JAR is:
\nJ’ (-4, 5), A’ (-6, 4), and R (-5, 2)
\nHence,f rom the above,
\nWe can conclude that option D matches with the given vertices of the reflection i.e., \u0394J’A’R’<\/p>\n

PART B
\nThe measure of \u2220A = 90\u00b0. What is m\u2220A’?
\nAnswer:
\nWe know that,
\nIn Reflection,
\nThe side lengths and the angle measures of the preimage and the image are the same
\nHence, from the above,
\nWe can conclude that
\n\u2220A = \u2220A’ = 90\u00b0<\/p>\n

Lesson 6.3 Analyze Rotations<\/h3>\n

Explain It!<\/strong>
\nMaria boards a car at the bottom of the Ferris wheel. She rides to the top, where the car stops. Maria tells her friend that she completed \\(\\frac{1}{4}\\) of the turn before the car stopped.
\n\"Envision
\nI can… rotate a two-dimensional figure.<\/p>\n

A. Do you agree with Maria? Explain.
\nAnswer:
\nIt is given that
\nMaria boards a car at the bottom of the Ferris wheel. She rides to the top, where the car stops. Maria tells her friend that she completed \\(\\frac{1}{4}\\) of the turn before the car stopped.
\nNow,
\nWe know that,
\nFrom the given figure,
\nThe Ferris wheel looks like a circle
\nWe know that
\nThe total angle measure of a circle (Ferris wheel) is: 360\u00b0
\nBut,
\nIt is given that
\nMaria tells her friend that she completed \\(\\frac{1}{4}\\) of the turn before the car stopped
\nBut, from the figure,
\nWe can observe that she completed the \\(\\frac{1}{2}\\) of the ride
\nHence, from the above,
\nWe can conclude that we can’t agree with Maria<\/p>\n

B. How could you use angle measures to describe the change in position of the car?
\nAnswer:
\nWe know that,
\nFrom the given figure,
\nThe Ferris wheel looks like a circle
\nWe know that
\nThe total angle measure of a circle (Ferris wheel) is: 360\u00b0
\nSo,
\nThe starting position of the car in terms of angle measure is given as 0\u00b0
\nThe \\(\\frac{1}{2}\\) of the position of the car in terms of angle measure = \\(\\frac{360\u00b0}{2}\\)
\n= 180\u00b0
\nHence, from the above,
\nWe can conclude that
\nIn terms of angle measures,
\nThe starting position of the car is: 0\u00b0
\nThe \\(\\frac{1}{2}\\) of the car is: 180\u00b0<\/p>\n

Focus on math practices
\nConstruct Arguments How can you describe Maria’s change in position when her car returns to the position at which she began the ride?
\nAnswer:
\nWe know that,
\nThe starting position of the car is: 0\u00b0
\nThe \\(\\frac{1}{2}\\) of the car is: 180\u00b0
\nNow,
\nIf Maria’s car returns to the position at which she began the ride, then
\nMaria has completed the Ferris wheel (or) Maria returned back from her previous \\(\\frac{1}{2}\\) of the position of the car
\nHence,
\nIn terms of angle measures,
\nThe change in Maria’s position when her car returns to the position at which she began the ride is: 360\u00b0 (or) -180\u00b0
\nHere,
\n-180\u00b0 represents that Maria returned back from her previous \\(\\frac{1}{2}\\) of the position of the car<\/p>\n

Essential Question<\/strong>
\nHow does rotation affect the properties of a two-dimensional figure?
\nAnswer:
\nWhen you rotate a two-dimensional figure, you are just moving it.
\nEx:
\nIf you rotate a rectangle, then it will remain a rectangle, just moved wherever you move it. This is similarly the same with an angle and aside length, the measure of the angle and the side length won’t change.<\/p>\n

Try It!<\/strong><\/p>\n

The architect continues to rotate the umbrella in a counterclockwise direction until it is in its original position. What is the angle of this rotation?
\n\"\"
\nAnswer:
\nIt is given that
\nThe architect continues to rotate the umbrella in a counterclockwise direction until it is in its original position.
\nNow,
\nWe know that,
\nIf any figure rotates until it comes to its original position again, then that means the figure completed a full cycle (or) revolution and the angle of the full cycle is: 360\u00b0
\nHence, from the above,
\nWe can conclude that the architect continues to rotate the umbrella for 360\u00b0 in a counterclockwise direction until it is in its original position<\/p>\n

Convince Me!
\nHow does an image compare to its preimage after a -45\u00b0 rotation?
\nAnswer:<\/p>\n

Try It!<\/strong><\/p>\n

The coordinates of the vertices of quadrilateral HIJK are H(1,4), I(3, 2), J(-1,-4), and K(-3, -2). If quadrilateral HIJK is rotated 270\u00b0 about the origin, what are the vertices of the resulting image, quadrilateral H’I’I’K?
\nAnswer:
\nIt is given that
\nThe coordinates of the vertices of quadrilateral HIJK are H(1,4), I(3, 2), J(-1,-4), and K(-3, -2)
\nNow,.
\nWe know that,
\nThe change in the x and y-coordinates for the given angle of rotation is:
\n\"\"
\nNow,
\nIt is given that to rotate the quadrilateral HIJK 270\u00b0 about the origin to form the quadrilateral H’I’J’K’
\nSo,
\nThe vertices of the quadrilateral H’I’J’K’ are:
\nH’ (4, -1), I’ (2, -3), J’ (-4, 1), K’ (-2, 3)
\nHence,
\nThe representation of the quadrilateral HIJK and its image H’I’J’K’ is:
\n\"\"<\/p>\n

Try It!<\/strong><\/p>\n

Describe the rotation that maps \u2206FGH to \u2206FG’H’.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of \u0394FGH are:
\nF (-5, -2), G (-1, -2), and H (-1, 1)
\nThe vertices of \u0394F’G’H’ are:
\nF’ (5, 2), G’ (1, 2), and H’ (1, -1)
\nNow,
\nTo find the rotation that maps \u0394FGH through \u0394F’G’H’, the following steps are:
\nStep 1:
\nDraw rays from the origin through point G and point G’
\nStep 2:
\nMeasure the angle formed by the rays
\nSo,
\nThe representation of \u0394FGH and its image \u0394F’G’H’ with its angle of rotation is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the angle of rotation that maps \u0394FGH through \u0394F’G’H’ is: 180\u00b0<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

A rotation is a transformation that turns a figure about a fixed point called the center of rotation. The angle of rotation is the number of degrees the figure is rotated. The x- and y-coordinates change in predictable ways when rotated.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How does rotation affect the properties of a two-dimensional figure?
\nAnswer:
\nWhen you rotate a two-dimensional figure, you are just moving it.
\nEx:
\nIf you rotate a rectangle, then it will remain a rectangle, just moved wherever you move it. This is similarly the same with an angle and aside length, the measure of the angle and the side length won’t change.<\/p>\n

Question 2.
\nReasoning<\/strong> If a preimage is rotated 360 degrees about the origin how can you describe its image?
\nAnswer:
\nWe know that,
\nTo complete the full cycle i.e., to a preimage to return to its original position, the angle measure is: 360\u00b0
\nHence, from the above,
\nWe can conclude that
\nIf a preimage is rotated 360\u00b0 about the origin, then the image and the preimage are the same<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> In Example 3, side AB is parallel to side DC. How are side A’ B’ and side D’ C’ related? Explain.
\n\"\"
\nAnswer:
\nIn Example 3,
\nThe given two-dimensional figure is a parallelogram
\nWe know that,
\nIn a parallelogram, the opposite sides have the same side lengths
\nSo,
\nNow,
\nIt is given that
\nIn parallelogram ABCD,
\nAB is parallel to CD
\nSo,
\nIn parallelogram A’B’C’D’,
\nA’B’ is parallel to C’D’ since these two sides are just the images of the sides AB and CD
\nHence, from the above,
\nWe can conclude that A’B’ is parallel to C’D’<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nThe coordinates of the vertices of rectangle ABCD are A(3,-2), B(3, 2), C(-3, 2), and D(-3,-2).
\na. Rectangle ABCD is rotated 90\u00b0 about the origin. What are the coordinates of the vertices of rectangle A’B’C’D’?
\nAnswer:
\nIt is given that
\nThe coordinates of the vertices of rectangle ABCD are A(3,-2), B(3, 2), C(-3, 2), and D(-3,-2).
\nNow,
\nWe know that,
\nThe change in the x and y-coordinates for the given angle of rotation is:
\n\"\"
\nNow,
\nIt is given that to rotate the rectangle ABCD 90\u00b0 about the origin to form the rectangle A’B’C’D’
\nSo,
\nThe vertices of the rectangle A’B’C’D’ are:
\nA’ (2, 3), B’ (-2, 3), C’ (-2, -3), and D’ (2, -3)
\nHence,
\nThe representation of the rectangle ABCD and its image rectangle A’B’C’D’ is:
\n\"\"<\/p>\n

b. What are the measures of the angles of A’B’C’D’?
\nAnswer:
\nFrom part (a),
\nWe can observe that all the angles of the rectangle A’B’C’D’ are the same
\nHence, from the above,
\nWe can conclude that the angle measures of the rectangle A’B’C’D’ are:
\n\u2220A = 90\u00b0,\u00a0\u2220B = 90\u00b0, \u2220C = 90\u00b0,and \u2220D = 90\u00b0<\/p>\n

Question 5.
\nDescribe the counterclockwise rotation that maps \u2206QRS to \u2206Q’R’S.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of \u0394QRS are:
\nQ (6, 2), R (4, 9), and S (2, 7)
\nThe vertices of \u0394Q’R’S’ are:
\nQ’ (2, -6), R’ (9, -4), and S’ (7, -2)
\nNow,
\nTo find the rotation that maps \u0394QRS through \u0394Q’R’S’, the following steps are:
\nStep 1:
\nDraw rays from the origin through point Q and point Q’
\nStep 2:
\nMeasure the angle formed by the rays
\nSo,
\nThe representation of \u0394QRS and its image \u0394Q’R’S’ with its angle of rotation is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the angle of rotation that maps \u0394QRS through \u0394Q’R’S’ is: 90\u00b0<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 6.
\nWhat is the angle of rotation about the origin that maps \u25b3\u03a1QR to \u25b3\u03a1’ \u039f’ R’?
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of \u0394PQR are:
\nP (3, 3), Q (5, 3), and R (5, 7)
\nThe vertices of \u0394P’Q’R’ are:
\nP’ (-3, 3), Q’ (-3, 5), and S’ (-7, 5)
\nNow,
\nTo find the rotation that maps \u0394PQR through \u0394P’Q’R’, the following steps are:
\nStep 1:
\nDraw rays from the origin through point P and point P’
\nStep 2:
\nMeasure the angle formed by the rays
\nSo,
\nThe representation of \u0394PQR and its image \u0394P’Q’R” with its angle of rotation is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the angle of rotation that maps \u0394PQR through \u0394P’Q’R’ is: 90\u00b0<\/p>\n

Question 7.
\nIs \u25b3X’ Y’Z’ a rotation of \u25b3XYZ? Explain.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394XYZ are:
\nX (-2, 3), Y (-5, 4), and Z (-2, 7)
\nThe vertices of \u0394X’Y’Z’ are:
\nX’ (3, 2), Y’ (4, 5), and Z’ (7, 2)
\nNow,
\nTo find the angle of rotation,
\nCompare the x and y-coordinates of \u0394XYZ and \u0394X’Y’Z’
\nSo,
\n(x, y) before rotation ——-> (y, -x) after rotation
\nWe know that,
\n\"\"
\nHence, from the above,
\nWe can conclude that \u0394X’Y’Z’ is a rotation of \u0394XYZ<\/p>\n

Question 8.
\n\u25b3PQR is rotated 270\u00b0 about the origin. Graph and label the coordinates of P’, Q’, and R’.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of \u0394PQR are:
\nP (2, 3), Q (4, 6), and R (2, 7)
\nIt is given that
\nRotate \u0394PQR 270\u00b0 at the origin
\nWe know that,
\n\"\"
\nSo,
\nThe vertices of \u0394P’Q’R’ when the angle of rotation is 270\u00b0 are:
\nP’ (3, -2), Q’ (6, -4), and R’ (7, -2)
\nHence,
\nThe representation of \u0394PQR and its image \u0394P’Q’R’ is:
\n\"\"<\/p>\n

Question 9.
\nIs \u25b3P’ Q’R’a 270\u00b0 rotation of \u25b3PQR about the origin? Explain.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394PQR are:
\nP (3, 3), Q (7, 4), and R (3, 5)
\nThe vertices of \u0394P’Q’R’ are:
\nP’ (-3, 3), Q’ (-4, 7), and R’ (-5, 3)
\nNow,
\nTo find the angle of rotation,
\nCompare the x and y-coordinates of \u0394PQR and \u0394P’Q’R’
\nSo,
\n(x, y) before rotation ——-> (-y, x) after rotation
\nWe know that,
\n\"\"
\nHence, from the above,
\nWe can conclude that \u0394P’Q’R’ is a rotation of \u0394PQR<\/p>\n

Question 10.
\nReasoning Explain why any rotation can be described by an angle between 0\u00b0 and 360\u00b0.
\nAnswer:
\nIf you rotate an object 360\u00b0, it\u2019s like the object never moved because the object would still be in the same spot as if you didn\u2019t move it.
\nHence,
\nAny rotation can be described by an angle of 0\u00b0 to 360\u00b0<\/p>\n

Question 11.
\nRotate rectangle KLMN 270\u00b0 about the origin.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of rectangle KLMN are:
\nK (-3, 2), L (-5, 2), M (-5, 4), and N (-3, 4)
\nIt is given that
\nRotate rectangle KLMN 270\u00b0 at the origin
\nWe know that,
\n\"\"
\nSo,
\nThe vertices of rectangle K’L’M’N’ when the angle of rotation is 270\u00b0 are:
\nK’ (2, 3), L’ (2, 5), M’ (4, 5), and N’ (4, 3)
\nHence,
\nThe representation of rectangle KLMN and its image rectangle K’L’M’N’ is:
\n\"\"<\/p>\n

 <\/p>\n

Question 12.
\nHigher-Order Thinking<\/strong> An architect is designing a new windmill with four sails. In her sketch, the sails’ center of rotation is the origin, (0, 0), and the tip of one of the sails, point Q, has coordinates (2, -3). She wants to make another sketch that shows the windmill after the sails have rotated 270\u00b0 about their center of rotation. What would be the coordinates of?
\nAnswer:
\nIt is given that
\nAn architect is designing a new windmill with four sails. In her sketch, the sails’ center of rotation is the origin, (0, 0), and the tip of one of the sails, point Q, has coordinates (2, -3). She wants to make another sketch that shows the windmill after the sails have rotated 270\u00b0 about their center of rotation
\nSo,
\nWe have to rotate point Q 270\u00b0 about the origin
\nWe know that,
\n\"\"
\nSo,
\nWhen we rotate any point 270\u00b0 about the origin,
\n(x, y) before rotation ——–> (y, -x)
\nSo,
\nThe representation of point Q after representation is: (-3, -2)
\nHence,
\nThe representation of point Q and its image Q’ is:
\n\"\"<\/p>\n

Assessment Practice<\/strong>
\nQuestion 13.
\nA rotation about the origin maps \u25b3TRI to \u25b3T’ R’I’.
\nPART A
\nWhich graph shows an angle you could measure to find the angle of rotation about the origin?
\n\"Envision
\nAnswer:
\nIt is given that
\nA rotation about the origin maps \u25b3TRI to \u25b3T’ R’I’
\nNow,
\nWe know that,
\nWe can find the angle of rotation only when the vertex maps with its image but not with any other images
\nHence, from the above,
\nWe can conclude that option A matches the given situation<\/p>\n

PART B
\nWhat is the angle of rotation about the origin?
\nA. 90\u00b0
\nB. 180\u00b0
\nC. 270\u00b0
\nD. 360\u00b0
\nAnswer:
\nFrom part (a),
\nWhen we observe option A,
\nThe vertices of \u0394TRI are:
\nT (-3, 3), R (-5, 3), and I (-4, 5)
\nThe vertices of \u0394T’R’I’ are:
\nT’ (3, -3), R’ (5, -3), and I’ (4, -5)
\nWe know that,
\n\"\"
\nSo,
\nWhen we compare the vertices of \u0394TRI and \u0394T’R’I’,
\nWe can observe that
\n(x, y) before rotation ———–> (-x, -y) after rotation
\nHence, from the above,
\nWe can conclude that the angle of rotation about the origin is: 180\u00b0<\/p>\n

Lesson 6.4 Compose Transformations<\/h3>\n

Solve & Discuss It!<\/strong>
\nHow can you map Figure A onto Figure B?
\n\"Envision
\nI can… describe and perform a sequence of transformations.
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nThe steps to obtain Figure B from Figure A are:
\nStep 1:
\nReflect Figure A across the x-axis
\nStep 2:
\nReflect the figure that we obtained in Step 1 across the y-axis
\nHence,
\nThe representation of the mapping of Figure A and Figure B using the above steps is:
\n\"\"<\/p>\n

Focus on math practices
\nLook for Relationships Is there another transformation or sequence of transformations that will map Figure A to Figure B?
\nAnswer:
\nYes, there are another sequence of transformations that will map Figure A to figure B
\nNow,
\nThe steps to obtain Figure B from Figure A are:
\nStep 1:
\nReflect Figure A across the y-axis
\nStep 2:
\nReflect the figure that we obtained in Step 1 across the x-axis
\nHence,
\nThe representation of the mapping of Figure A and Figure B using the above steps is:
\n\"\"<\/p>\n

Essential Question<\/strong>
\nHow can you use a sequence of transformations to map a preimage to its image?
\nAnswer:
\nMathematical\u00a0transformations\u00a0involve changing an image in some prescribed manner. There are\u00a0four\u00a0main\u00a0types of transformations They are:
\nA) Translation B) Rotation C) Reflection D) Dilation<\/p>\n

Try It!<\/strong><\/p>\n

Ava decided to move the cabinet to the opposite wall. What sequence of transformations moves the cabinet to its new position?
\n\"\"
\nAnswer:
\nIt is given that
\nAva decided to move the cabinet to the opposite wall
\nSo,
\nFrom the figure,
\nWe can observe that
\nTo move the cabinet wall to the opposite wall, the following sequences of transformations have to be followed:
\nStep 1:
\nTranslate the cabinet 8 units down
\nStep 2:
\nRotate the cabinet 360\u00b0 counterclockwise
\nHence, from the above,
\nWe can conclude that
\nThe new position of the cabinet is:
\n\"\"<\/p>\n

Convince Me!
\nAva decides that she would like the chairs to be placed directly across from the couch. What is a sequence of transformations that she can use to move the chairs to their new positions?<\/p>\n

Try It!<\/strong><\/p>\n

What is another sequence of transformations that maps \u25b3ABC onto \u25b3A” B” C”?
\n\"\"
\nAnswer:
\nThe given coordinate plane is:
\n\"\"
\nFrom the given coordinate plane,
\nThe vertices of \u0394ABC are:
\nA (-5, 5), B (-3, 3), and C (-6, 1)
\nNow,
\nAnother sequence of transformations that maps \u0394ABC onto \u0394A”B” C” is:
\nStep 1:
\nDraw \u0394ABC and make its reflection across the y-axis and name it \u0394A’B’C’
\nStep 2:
\nTranslate \u0394A’B’C’ 2 units to the right and 6 units down
\nStep 3:
\nReflect the figure we obtained in step 2 across the y-axis
\nHence, from the above,
\nWe can conclude that
\nThe representation of another sequence of transformations that maps \u0394ABC onto \u0394A”B” C” is:
\n\"\"<\/p>\n

KEY CONCEPT<\/strong>
\nYou can use a sequence of two or more transformations to map a preimage to its image.
\nYou can map \u25b3ABC onto \u25b3\u0391” \u0392” C” by translation of 3 units right followed by a 90\u00b0 clockwise rotation about the origin.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How can you use a sequence of transformations to map a preimage to its image?
\nAnswer:
\nMathematical\u00a0transformations\u00a0involve changing an image in some prescribed manner. There are\u00a0four\u00a0main\u00a0types of transformations They are:
\nA) Translation B) Rotation C) Reflection D) Dilation<\/p>\n

Question 2.
\nMake Sense and Persevere A preimage is rotated 180\u00b0 about the origin and then rotated 180\u00b0 about the origin again. Compare the preimage and image.
\nAnswer:
\nIt is given that
\nA preimage is rotated 180\u00b0 about the origin and then rotated 180\u00b0 about the origin again
\nNow,
\nWhen the preimage is rotated 180\u00b0 and again 180\u00b0
\nThe image will be the same as the preimage
\nHence, from the above,
\nWe can conclude that
\nWhen a preimage is rotated 180\u00b0 about the origin and then rotated 180\u00b0 about the origin again, the preimage and the image will be the same<\/p>\n

Question 3.
\nReasoning<\/strong> A figure ABC, with vertices A(2, 1), B(7, 4), and C(2, 7), is rotated 90\u00b0 clockwise about the origin, and then reflected across the x-axis. Describe another sequence that would result in the same image.
\nAnswer:
\nIt is given that
\nA figure ABC, with vertices A(2, 1), B(7, 4), and C(2, 7), is rotated 90\u00b0 clockwise about the origin, and then reflected across the x-axis
\nSo,
\nThe steps for another sequence of transformations that would result in the same image as the given situation is:
\nStep 1:
\nDraw the given vertices of Triangle ABC
\nStep 2:
\nRotate Triangle ABC 90\u00b0 counterclockwise
\nStep 3:
\nReflect the image that we obtained in step 2 across the x-axis so that we will get the same image as in the given situation
\nHence,
\nThe representation of another sequence of transformations is:
\n\"\"<\/p>\n

Do You Know How?<\/strong>
\nIn 4-6, use the diagram below.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of Figure WXYZ are:
\nW (2, 4), X (5, 4), Y (5, 2), and Z (2, 2)
\nThe vertices of Figure W’X’Y’Z’ are:
\nW’ (-4, -4), X’ (-4, -1), Y’ (-2, -1), and Z’ (-2, -4)<\/p>\n

Question 4.
\nDescribe a sequence of transformations that maps rectangle WXYZ onto rectangle W’X’Y’Z’.
\nAnswer:
\nThe steps that we have to follow to obtain the given sequence of transformations that maps rectangle WXYZ onto rectangle W’X’Y’Z’ are:
\nStep 1:
\nDraw rectangle WXYZ
\nStep 2:
\nReflect rectangle WXYZ across the x-axis
\nStep 3:
\nRotate the image we obtained in step 2 90\u00b0 counterclockwise
\nStep 4:
\nTranslate the image we obtained in step-3 6 units left so that we can obtain rectangle W’X’Y’Z’
\nHence,
\nThe representation of the sequence of transformations for the given situation is:
\n\"\"<\/p>\n

Question 5.
\nDescribe another way that you could map rectangle WXYZ onto W’X’Y’Z’.
\nAnswer:
\nThe steps for another way of transformations that maps rectangle WXYZ onto W’X’Y’Z’ are:
\nStep 1:
\nDraw Rectangle WXYZ
\nStep 2:
\nRotate rectangle WXYZ 90\u00b0 counterclockwise
\nStep 3:
\nRotate the image we obtained in step-2 180\u00b0 counterclockwise
\nStep 4:
\nTranslate the image we obtained in step 3 6 units down so that we can obtain rectangle W’X’Y’Z’
\nHence,
\nThe representation of another sequence of transformations is:
\n\"\"<\/p>\n

Question 6.
\nDraw the image of rectangle WXYZ after a reflection across the line y = 1 and a translation 1 unit right. Label the image W” X” Y” Z”.
\nAnswer:
\nThe vertices of rectangle WXYZ are:
\nW (2, 4), X (5, 4), Y (5, 2), and Z (2, 2)
\nAfter the reflection across the line y = 1,
\nThe vertices of rectangle WXYZ are:
\nW’ (2, -5), X’ (5, -5), Y’ (5, -3), and Z’ (2, -3)
\nAfter the translation of 1 unit right,
\nThe vertices of rectangle WXYZ are:
\nW” (3, -5), X” (6, -5), Y” (6, -3), and Z” (3, -3)
\nHence,
\nThe representation of rectangle WXYZ and its image W”X” Y”Z” after the above sequence of transformations is:
\n\"\"<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nLeveled Practice Describe a sequence of transformations that maps \u25b3QRS onto \u25b3TUV.
\n\"Envision
\nA translation ________ units left and ________ units down, followed by a _________ across the ________.
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394QRS are:
\nQ (3, 4), R (7, 4), and S (6, 9)
\nThe vertices of \u0394TUV are:
\nT (0, 0), U (-4, 0), and V (-3, 5)
\nNow,
\nIn order to obtain the vertices of \u0394TUV, the following transformations we have to follow are:
\nStep 1:
\nTranslate the vertices of \u0394QRS 3 units left and 4 units down
\nStep 2:
\nReflect the image we obtained in step 1 across the x-axis in order to get the vertices of \u0394TUV
\nHence, from the above,
\nWe can conclude that
\nA translation of 3 units left and 4 units down, followed by a reflection across the x-axis<\/p>\n

Question 8.
\nModel with Math
\nA family moves a table, shown as rectangle EFGH, by translating it 3 units left and 3 units down followed by a 90\u00b0 rotation about the origin. Graph E’ F’G’H’ to show the new location of the table.
\n\"Envision
\nAnswer:
\nIt is given that
\nA family moves a table, shown as rectangle EFGH, by translating it 3 units left and 3 units down followed by a 90\u00b0 rotation about the origin
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of rectangle EFGH are:
\nE (3, 3), F (8, 3), G (8, 7), and H (3, 7)
\nNow,
\nTo obtain rectangle E’F’G’H’,
\nThe following series of transformations are:
\nStep 1:
\nBy translating 3 units left and 3 units down,
\nThe vertices of rectangle EFGH will become:
\nE (3 – 3, 3 – 3), F (8 – 3, 3 – 3), G (8 – 3, 7 – 3), and H (3 – 3, 7 – 3)
\nE (0, 0), F (5, 0), G (5, 4), and H (0, 4)
\nStep 2:
\nRotate the vertices we obtain in step 1 90\u00b0 counterclockwise about the origin
\nWe know that,
\nWhen we rotate a point 90\u00b0 counterclockwise about the origin,
\n(x, y) before rotation ——> (-y, x) after rotation
\nSo,
\nThe vertices of rectangle E’F’G’H’ are:
\nE’ (0, 0), F’ (0, 5), G’ (-4, 5), and H’ (-4, 0)
\nHence,
\nThe representation of the new location of the table is:
\n\"\"<\/p>\n

Question 9.
\nDescribe a sequence of transformations that maps quadrilateral ABCD to quadrilateral HIJK.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of quadrilateral ABCD are:
\nA (3, 1), B (4, 1), C (4, 3), and D (3, 3)
\nThe vertices of quadrilateral HIJK are:
\nH (-3, 0), I (-2, 0), J (-2, -2), and K (-3, -2)
\nNow,
\nThe series of transformations that maps quadrilateral ABCD onto quadrilateral HIJK are:
\nStep 1:
\nDraw quadrilateral ABCD
\nStep 2:
\nReflect quadrilateral ABCD across the x-axis
\nSo,
\nThe vertices of quadrilateral ABCD are:
\nA (3, -1), B (4, -1), C (4, -3), and D (3, -3)
\nStep 3:
\nTranslate 6 units left and 1 unit up
\nSo,
\nThe vertices that we obtained in step 2 will become (The vertices of quadrilateral HIJK):
\nH (-3, 0), I (-2, 0), J (-2, -2), and K (-3, -2)
\nHence,
\nThe representation of the series of transformations that map quadrilateral ABCD onto quadrilateral HIJK are:
\n\"\"<\/p>\n

Question 10.
\nMap \u25b3QRS to \u25b3Q’R’ S’with a reflection across the y-axis followed by a translation 6 units down.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394QRS are:
\nQ (-3, 5), R (-2, 4), and S (-5, 3)
\nNow,
\nThe series of transformations that maps \u0394QRS to \u0394Q’R’S’ as given above are:
\nStep 1:
\nReflect the vertices of \u0394QRS along the y-axis
\nSo,
\nQ (3, 5), R (2, 4), and S (5, 3)
\nStep 2:
\nTranslate the vertices that we obtain in step-1 6 units down so that we can obtain the vertices of \u0394Q’R’S’
\nSo,
\nQ’ (3, 5 – 6), R’ (2, 4 – 6), and S’ (5, 3 – 6)
\nQ’ (3, -1), R’ (2, -2), and S’ (5, -3)
\nHence,
\nThe representation of the series of transformations for the given situation is:
\n\"\"<\/p>\n

 <\/p>\n

Question 11.
\nHigher-Order Thinking<\/strong> A student says that he was rearranging furniture at home and he used a glide reflection to move a table with legs from one side of the room to the other. Will a glide reflection result in a functioning table? Explain.
\nAnswer:
\nIt is given that
\nA student says that he was rearranging furniture at home and he used a glide reflection to move a table with legs from one side of the room to the other.
\nWe know that,
\nA glide reflection is a\u00a0 sequence of translation and reflection
\nNow,
\nFrom the given situation,
\nWe can observe that the table is moving from one room to the other
\nSo,
\nThe “Translation” occurs
\nNow,
\nAfter moving he will rearrange the table in the room
\nSo,
\nA “Reflection” may take place
\nHence, from the above,
\nWe can conclude that a glide reflection result in a functioning table<\/p>\n

Assessment Practice<\/strong>
\nQuestion 12.
\nPART A
\nWhich sequence of transformations maps rectangle ABCD onto rectangle A’ B’C’D?
\n\"Envision
\nA. translation 6 units down, reflection across the x-axis
\nB. reflection across the x-axis, translation 6 units right
\nC. reflection across the x-axis, translation 6 units left
\nD. translation 6 units left, reflection across the y-axis
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of rectangle ABCD are:
\nA (2, -2), B (4, -2), C (4, -3), and D (2, -3)
\nThe vertices of rectangle A’B’C’D’ are:
\nA’ (-4, 2), B’ (-2, 2), C’ (-2, 3), and D’ (-4, 3)
\nSo,
\nThe sequence of transformations that maps rectangle ABCD to rectangle A’B’C’D’ is:
\n\"\"
\nHence, from the above,
\nWe can conclude that option C matches the given situation<\/p>\n

PART B
\nDescribe a sequence of transformations that maps A’B’C’ D’ onto ABCD.
\nAnswer:
\nThe sequence of transformations that maps A’B’C’D’ onto ABCD is:
\nStep 1:
\nDraw the rectangle ABCD
\nSo,
\nThe vertices of rectangle ABCD are:
\nA (2, -2), B (4, -2), C (4, -3), and D (2, -3)
\nStep 2:
\nReflect rectangle ABCD across the x-axis
\nSo,
\nThe vertices of rectangle ABCD are:
\nA (2, 2), B (4, 2), C (4, 3), and D (2, 3)
\nStep 3:
\nTranslate the vertices that we obtained in step 2 6 units left so that we can obtain rectangle A’B’C’D’
\nSo,
\nA’ (2 – 6, 2), B’ (4 – 6, 2), C’ (4 – 6, 3), and D’ (2 – 6, 3)
\nSo,
\nA’ (-4, 2), B’ (-2, 2), C’ (-2, 3), and D’ (-4, 3)<\/p>\n

Question 13.
\nPART A
\nWhich figures are the image of Figure A after a reflection across the x-axis and a translation of 4 units right?
\n\"Envision
\nA. Figure B
\nB. Figure C
\nC. Figure D
\nD. Figure E
\nAnswer:
\nWe know that,
\nA “Reflection” is called a “Flip” but the reflection does not affect the shape and length of the figure
\nSo,
\nFrom the given figures,
\nWhen we observe Figures A and B,
\nWe can say that Figure B is a reflection of A because the shape and length is the same
\nBut,
\nwhen we observe the other figures,
\nWe have different shapes and lengths between the preimage and the image
\nHence, from the above,
\nWe can conclude that
\nFigure B is the image of Figure A after a reflection across the x-axis and a translation of 4 units right<\/p>\n

PART B
\nWhich figure can be transformed into Figure G after a rotation 90\u00b0 about the origin, then a translation 13 units right and 4 units down?
\nA. Figure B
\nB. Figure D
\nC. Figure E
\nD. Figure F
\nAnswer:
\nFrom the given figures,
\nWe have to obtain
\nThe coordinates of Figure G are:
\n(6, -6), (9, -6), (9, -9), (7, -9), and (6, -5)
\nWe know that,
\nWhen a point rotates 90\u00b0 about the origin,
\n(x, y) before rotation ——-> (-y, x) after rotation
\nNow,
\nFor the translation of 13 units right and 4 units down,
\nThe vertices will be like: (x + 13, y – 4)
\nHence, from the above,
\nWe can conclude that option A matches the given situation<\/p>\n

3-ACT MATH<\/strong><\/p>\n

3-Act Mathematical Modeling: Tricks of the Trade
\n\"Envision<\/p>\n

ACT 1<\/strong>
\nQuestion 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:<\/p>\n

Question 2.
\nWrite the Main Question you will answer.
\n\"Envision
\nAnswer:<\/p>\n

Question 3.
\nMake a prediction to answer this Main Question.
\nAnswer:<\/p>\n

Question 4.
\nConstruct Arguments Explain how you arrived at your prediction.
\nAnswer:<\/p>\n

ACT 2<\/strong>
\nQuestion 5.
\nWhat information in this situation would be helpful to know? How would you use that information?
\n\"Envision
\nAnswer:<\/p>\n

Question 6.
\nUse Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
\nAnswer:<\/p>\n

Question 7.
\nModel with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 8.
\nWhat is your answer to the Main Question? Does it differ from your prediction? Explain.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong>
\nQuestion 9.
\nWrite the answer you saw in the video.
\n\"Envision
\nAnswer:<\/p>\n

Question 10.
\nReasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\nAnswer:<\/p>\n

Question 11.
\nMake Sense and Persevere Would you change your model now that you know the answer? Explain.
\nAnswer:<\/p>\n

ACT 3 Extension<\/strong>
\nReflect
\nQuestion 12.
\nModel with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\n\"Envision
\nAnswer:<\/p>\n

Question 13.
\nMake Sense and Persevere When did you struggle most while solving the problem? How did you overcome that obstacle?
\nAnswer:<\/p>\n

SEQUEL
\nQuestion 14.
\nBe Precise Find another optical illusion online involving shapes that look different but are the same. Explain how you know the shapes are the same.
\n\"Envision
\nAnswer:<\/p>\n

Lesson 6.5 Understand Congruent Figures<\/h3>\n

Solve & Discuss It!<\/strong>
\nSimone plays a video game in which she moves shapes into empty spaces. After several rounds, her next move must fit the blue piece into the dashed space. How can Simone move the blue piece to fit in the space?
\n\"Envision
\nI can… use a sequence of translations, reflections, and rotations to show that figures are congruent.
\nAnswer:
\nIt is given that
\nSimone plays a video game in which she moves shapes into empty spaces. After several rounds, her next move must fit the blue piece into the dashed space.
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe blue piece and the dashed\u00a0 piece are the reflections of each other
\nHence,
\nSimone move the blue piece to fit in the space by the translation followed by reflection<\/p>\n

Reasoning
\nHow can you use what you know about sequences of transformations to move the piece?
\nAnswer:
\nFrom the given figure,
\nWe can observe that
\nThe blue piece should be moved to the place of the dashed space and it will be possible only due to the “Translation”
\nBut,
\nWe can observe that it does not fit into the dashed space.
\nSo,
\nReflect the blue piece that it can fit into the dashed piece
\nHence, from the above,
\nWe can conclude that the sequence of transformations we can use to move the transformation are:
\na. Translation b. Reflection<\/p>\n

Focus on math practices
\nConstruct Arguments How do you know that the piece that fits into the space is the same as the original blue shape? Explain.
\nAnswer:
\nWe know that,
\nIn the reflection,
\na. The shape of the preimage and the image are the same
\nb. The length of the preimage and the image are the same
\nc. The orientation of the image and the preimage are different
\nHence, from the above,
\nWe can conclude that
\nDue to the properties of the reflection,
\nWe know that the piece that fits into space is the same as the original shape<\/p>\n

Essential Question<\/strong>
\nHow does a sequence of translations, reflections, and rotations result in congruent figures?
\nAnswer:
\nIf we copy\u00a0one figure\u00a0on tracing paper and move the paper so the copy covers\u00a0the other figure\u00a0exactly, then that suggests they are\u00a0congruent. We can prove that two\u00a0figures\u00a0are\u00a0congruent\u00a0by describing a\u00a0sequence\u00a0of translations, rotations, and reflections that move\u00a0one figure\u00a0onto\u00a0the other\u00a0so they match up exactly.<\/p>\n

Try It!<\/strong><\/p>\n

How can you determine whether the orange and blue rectangles are congruent?
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"\"
\nFrom the given coordinate plane,
\nWe can observe that
\nThe blue rectangle has a length of 6 units and a width of 5 units
\nThe orange rectangle has a length of 5 units and a width of 6 units
\nNow,
\nWe can say that
\nWe can obtain the orange rectangle by rotating the blue rectangle
\nWe know that,
\nRotations, reflections, and translations are isometric. That means that these transformations do not change the size of the figure.\u00a0If the size and shape of the figure is not changed, then the figures are congruent
\nHence, from the above,
\nWe can conclude that the orange and blue rectangles are congruent<\/p>\n

Convince Me!
\nQuadrilateral PQRS is congruent to quadrilateral P’ Q’R’S. What do you know about how these figures relate?
\nAnswer:
\nIt is given that
\nQuadrilateral PQRS is congruent to quadrilateral P’ Q’R’S
\nWe know that,
\nWhen the two figures are congruent,
\na. The shapes of the two figures are the same
\nb. The lengths of the two figures are the same
\nc. The angle measures of the two figures are the same
\nHence,
\nIn Quadrilateral PQRS and Quadrilateral P’Q’R’S’, the two figures are said to be congruent when
\na. PQ = P’Q’ and RS = R’S’
\nb. \u2220P = \u2220P’, \u2220Q = \u2220Q’, \u2220R = \u2220R’, and \u2220S = \u2220S’<\/p>\n

Try It!<\/strong><\/p>\n

Are the figures congruent? Explain.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of Figure 1 are:
\n(1, 5), (3, 7), (3, 5), and (2, 3)
\nThe vertices of Figure 2 are:
\n(6, 3), (8, 3), (10, 2), and (8, 1)
\nNow,
\nFind out whether Translation, Reflection, and Rotation is possible between 2 figures or not
\nSo,
\nThe representation of Figure 1 and Figure 2 are:
\n\"\"
\nNow,
\nFrom the given figures,
\nWe can observe that none of the transformations is possible
\nHence, from the above,
\nWe can conclude that the two figures are not congruent<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

Two-dimensional figures are congruent if there is a sequence of translations, reflections, and rotations that maps one figure onto the other.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How does a sequence of translations, reflections, and rotations result in congruent figures?
\nAnswer:
\nIf we copy\u00a0one figure\u00a0on tracing paper and move the paper so the copy covers\u00a0the other figure\u00a0exactly, then that suggests they are\u00a0congruent. We can prove that two\u00a0figures\u00a0are\u00a0congruent\u00a0by describing a\u00a0sequence\u00a0of translations, rotations, and reflections that move\u00a0one figure\u00a0onto\u00a0the other\u00a0so they match up exactly.<\/p>\n

Question 2.
\nReasoning<\/strong> Does a sequence of transformations have to include a translation, a reflection, and a rotation to result in congruent figures? Explain.
\nAnswer:
\nRotations,\u00a0reflections, and\u00a0translations\u00a0are isometric. That means that these\u00a0transformations do\u00a0not change the size of the\u00a0figure. If the size and\u00a0shape\u00a0of the\u00a0figure\u00a0is not changed, then the\u00a0figures\u00a0are\u00a0congruent<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Is there a sequence of reflections, rotations, and translations that makes the preimage and image not only congruent, but identical in orientation? Explain.
\nAnswer:
\nWe know that,
\nIn the reflection,
\nThe orientation of the preimage and the image will differ
\nHence, from the above,
\nWe can conclude that
\nThere is a sequence of reflections, rotations, and translations that makes the preimage and image only congruent but not identical in orientation<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nA rectangle with an area of 25 square centimeters is rotated and reflected in the coordinate plane. What will be the area of the resulting image? Explain.
\nAnswer:
\nIt is given that
\nA rectangle with an area of 25 square centimeters is rotated and reflected in the coordinate plane.
\nWe know that,
\nIn a sequence of transformations like Translation, Rotation, and Reflection,
\nThe shapes and side lengths of the image and the preimage are the same
\nSince the side lengths are the same, the area will also be the same
\nHence, from the above,
\nWe can conclude that the area of the image will also be 25 square centimeters<\/p>\n

In 5 and 6, use the coordinate grid below.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394ABC are:
\nA (1, 4), B (2, 2), and C (5, 2)
\nThe vertices of \u0394DEF are:
\nD (9, 9), E (8, 7), and F (5, 7)
\nThe vertices of \u0394GHI are:
\nG (6, 6), H (8, 5), and I (8, 1)<\/p>\n

Question 5.
\nIs \u25b3ABC \u2245 \u25b3DEF? Explain.
\nAnswer:
\nTo find whether \u0394ABC is congruent to \u0394DEF or not,
\nStep 1:
\nReflect Triangle ABC across the x-axis
\nStep 2:
\nTranslate the image that we obtained in step-1 10 units right and 5 units up
\nSo,
\nThe representation of step 1 and step 2 is:
\n\"\"
\nHence, from the above,
\nWe can observe that the vertices we obtained in step 2 are the same as \u0394DEF
\nHence,
\n\u0394ABC is congruent to \u0394DEF<\/p>\n

Question 6.
\nIs \u25b3ABC \u2245 \u25b3GHI? Explain.
\nAnswer:
\nThe representation of \u25b3ABC and \u25b3GHI is:
\n\"\"
\nFrom the above,
\nWe can observe that \u25b3ABC and \u25b3GHI do not have the same size
\nWe know that,
\nIn order to be 2 figures congruent,
\na. The sizes of the figures would be the same
\nb. The shapes of the figures would be the same
\nc. The side lengths of the figures should be the same
\nHence, from the above,
\nWe can conclude that
\n\u25b3ABC is not congruent to \u25b3GHI<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\n\u25b3Q’R’ S’ is the image of \u25b3QRS after a reflection across the y-axis and a translation 6 units down. Is the image the same size and shape as the preimage?
\n\"Envision
\n\u25b3QRS and \u25b3Q’R’S’ _________ the same size and shape.
\nAnswer:
\nIt is given that
\n\u25b3Q’R’ S’ is the image of \u25b3QRS after a reflection across the y-axis and a translation 6 units down
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394QRS are:
\nQ (-3, 5), R (-2, 4), and S (-5, 3)
\nThe vertices of \u0394Q’R’S’ are:
\nQ’ (3, -1), R’ (2, -2), and S’ (5, -3)
\nNow,
\nStep 1:
\nAfter a reflection of \u0394QRS across the y-axis,
\nThe vertices of \u0394QRS will be:
\nQ (3, 5), R (2, 4), and S (5, 3)
\nStep 2:
\nAfter a translation of 6 units down,
\nThe vertices that we obtained in Step 1 are:
\nQ’ (3, 5 – 6), R’ (2, 4 – 6), and S’ (5, 3 – 6)
\nQ’ (3, -1), R’ (2, -2), and S’ (5, -3)
\nSo,
\nThe vertices of \u0394Q’R’S’ we obtained from the coordinate plane and the vertices of \u0394Q’R’S’ we obtained after the sequence of transformations are the same
\nHence, from the above,
\nWe can conclude that \u25b3QRS and \u25b3Q’R’S’ have the same size and shape and the image (\u0394Q’R’S’) is the same size and shape as the preimage (\u0394QRS)<\/p>\n

Question 8.
\nIs \u25b3DEF \u2245 \u25b3D’ E’F’? Explain.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394DEF are:
\nD (5, 5), E (6, 3), and F (2, 4)
\nThe vertices of \u0394D’E’F’ are:
\nD’ (-2, -1), E’ (-3, 1), and F’ (1, 0)
\nSo,
\nThe representtaion of the sequence of transformations of \u0394DEF to show it is congruent to \u0394D’E’F’ is:
\n\"\"
\nHence, from the above,
\nWe can conclude that \u0394DEF is congruent to \u0394D’E’F’<\/p>\n

Question 9.
\nConstruct Arguments Describe a way to show that quadrilateral ABCD is congruent to quadrilateral A’B’C’D’.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of quadrilateral ABCD are:
\nA (3, 5), B (5, 5), C (5, 4), and D (3, 4)
\nThe vertices of quadriateral A’B’C’D’ are:
\nA’ (-3, 0), B’ (-5, 0), C’ (-5, -1), and D’ (-3, -1)
\nSo,
\nThe representtaion of the transformation of sequences that shows quadrilateral ABCd is congruent to quadrilateral A’B’C’D’ is:
\n\"\"
\nHence, from the above,
\nWe can conclude that quadrilateral ABCD is congruent to quadrilateral A’B’C’D’<\/p>\n

Question 10.
\nYou are making two triangular flags for a project and need the flags to be the same shape and size. \u25b3XYZ and \u25b3X’Y’Z’ are the flags you have drawn. Are the flags the same shape and size? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nYou are making two triangular flags for a project and need the flags to be the same shape and size. \u25b3XYZ and \u25b3X’Y’Z’ are the flags you have drawn.
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordiate plane,
\nThe vertices of \u0394XYZ are:
\nX (5, 6), Y (6, 1), and Z (2, 2)
\nhe vertices of \u0394X’Y’Z’ are:
\nX’ (-2, 0), Y’ (-3, -5), and Z’ (1, -4)
\nSo,
\nThe representtaion of the sequence of transformation to find whether two flags have the same size and the same shape or not is:
\n\"\"<\/p>\n

Question 11.
\nWhich two triangles are congruent? Describe the sequence of transformations that maps one figure onto the other.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394ABC are:
\nA (-3, 2), B (-2, 7), and C (-7, 4)
\nThe vertices of \u0394QRS are:
\nQ (4, 7), R (2, 2), and S (7, 2)
\nThe vertices of \u0394XYZ are:
\nX (-2, -7), Y (-2, -3), and Z (-7, -5)
\nThe vertices of \u0394DEF are:
\nD (7, -5), E (2, -2), and F (2, -7)
\nNow,
\nWe know that,
\nThe two figures are said to be congruent when
\na. The 2 figures have the same size i.e., the same length and the same angle measure
\nb. The 2 figures have the same shape
\nSo,
\nThe figures that are congruent to each other are represented as:
\n\"\"
\nSo,
\nFrom the above,
\nWe can say that \u0394QRS and \u0394DEf are congruent
\nHence,
\nThe sequence of Transformations to show \u0394QRS and \u0394DEF are congruent is:
\n\"\"<\/p>\n

Question 12.
\nIs \u25b3LMN \u2245 \u25b3XYZ? Explain.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394LMN are:
\nL (7, 9), M (9, 5), and N (6, 5)
\nThe vertices of \u0394XYZ are:
\nX (2, 2), Y (5, 4), and Z (5, 1)
\nNow,
\nWe know that,
\nThe two figures are said to be congruent when
\na. The 2 figures have the same size i.e., the same length and the same angle measure
\nb. The 2 figures have the same shape
\nSo,
\nThe sequence of transformations to find whether \u0394LMN is congruent to \u0394XYZ is:
\n\"\"
\nHence, from the above,
\nWe can conclude that \u0394LMN is not congruent to \u0394XYZ since they don’t have the same side lengths<\/p>\n

Question 13.
\nHigher-Order Thinking<\/strong> A student was asked to describe a sequence of transformations that maps \u25b3DEF onto \u25b3D’ E’F’, given that \u25b3DEF \u2245 \u25b3D’ E’F’. She incorrectly said the sequence of transformations that maps \u25b3DEF onto \u25b3D’ E’F’ is a reflection across the x-axis, followed by a translation of 6 units right and 4 units up. What mistake did the student likely make?
\n\"Envision
\nAnswer:
\nIt is given that
\nA student was asked to describe a sequence of transformations that maps \u25b3DEF onto \u25b3D’ E’F’, given that \u25b3DEF \u2245 \u25b3D’ E’F’. She incorrectly said the sequence of transformations that maps \u25b3DEF onto \u25b3D’ E’F’ is a reflection across the x-axis, followed by a translation of 6 units right and 4 units up
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394DEF are:
\nD (4, 5), E (5, 1), and F (1, 2)
\nThe vertices of \u0394D’E’F’ are:
\nD’ (-2, -1), E’ (-1, 3), and F’ (-5, 2)
\nNow,
\nThe correct sequence of Transformations to show \u0394DEF and \u0394D’E’F’ are congruent is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the mistake done by a student is the interchange of the sequence of Transformations
\nHence,
\nThe correct sequence of Transformations is:
\nTranslation of 6 units left and 4 units down followed by the reflection across the x-axis<\/p>\n

Assessment Practice<\/strong>
\nQuestion 14.
\nPART A
\nHow can you determine whether \u25b3DEF \u2245 \u25b3D’ E’ F?
\n\"Envision
\nA. Determine whether a sequence of rotations maps \u25b3DEF onto \u25b3D’E’F’.
\nB. Determine whether a sequence of transformations maps \u25b3DEF onto \u25b3D’ E’F’.
\nC. Determine whether a sequence of translations maps \u25b3DEF onto \u25b3D’ E’F’.
\nD. Determine whether a sequence of reflections maps \u25b3DEF onto \u25b3D’ E’F’.
\nAnswer:
\nWe know that,
\nIf we want to find whether the given figures are congruent or not, then
\nWe have to determine whether a sequence of transformations maps \u0394DEF onto \u0394D’E’F’
\nHence, from the above,
\nWe can conclude that option A matches the given situation<\/p>\n

PART B
\nIs \u25b3DEF \u2245 \u25b3D’ E’ F? Explain.
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394DEF are:
\nD (5, 5), E (6, 3), and F (2, 4)
\nThe vertices of \u0394D’E’F’ are:
\nD’ (-2, -1), E’ (-3, 1), and F’ (1, 0)
\nSo,
\nThe representation of the sequence of transformations to find out whether \u0394DEf and \u0394D’E’F’ are congruent or not is:
\n\"\"
\nHence, from the above,
\nWe can conclude that \u0394DEF is congruent to \u0394D’E’F’<\/p>\n

Topic 6 MID-TOPIC CHECKPOINT<\/h3>\n

Question 1.
\nVocabulary Describe three transformations where the image and preimage have the same size and shape. Lesson 6-1, Lesson 6-2, and Lesson 6-3
\nAnswer:
\nWe know that,
\nThere are four types of transformations. They are:
\na. Translation
\nb. Reflection
\nc. Rotation
\nd. Translation
\nNow,
\nSome transformations keep the pre-image and image congruent. Congruent means that they are the same size and shape or that they have the same measurements. They make not have the same orientation
\nHence,
\nThree of the four transformations that preserve the size and shape of the pre-image are: Translation, Rotation, and Reflections<\/p>\n

For 2-6, use the figures below.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of the quadrilateral MNPQ are:
\nM (1, 2), N (2, 4), P (4, 5), and Q (3, 3)
\nThe vertices of the quadrilateral RSTU are:
\nR (1, -2), S (2, -4), T (4, -5), and U (3, -3)<\/p>\n

Question 2.
\nWhat are the coordinates of each point after quadrilateral RSTU is rotated 90\u00b0 about the origin? Lesson 6-3
\nAnswer:
\nWe know that,
\nWhen any point is rotated 90\u00b0 about the origin,
\n(x, y) before rotation ——–> (-y, x) after rotation
\nHence,
\nAfter 90\u00b0 rotation about the origin,
\nThe coordinates of each point of the quadrilateral RSTU will become:
\nR (2, 1), S (4, 2), T (5, 4), and U (3, 3)<\/p>\n

Question 3.
\nWhat are the coordinates of each point after quadrilateral MNPQ is translated 2 units right and 5 units down? Lesson 6-1
\nAnswer:
\nWe know that,
\nWhen the translation occurs,
\n(x, y) before Translation ——-> (x + h, y + k) after Translation
\nWhere,
\nh is the translation on the x-axis
\nk is the translation on the y-axis
\nWe will take the positive value of h when the translation occurs on the right side of the x-axis
\nWe will take the negative value of h when the translation occurs on the left side of the x-axis
\nWe will take the positive value of k when the translation occurs on the top side of the y-axis
\nWe will take the negative value of k when the translation occurs on the down side of the y-axis
\nHence,
\nAfter the Translation of 2 units right and 5 units down,
\nThe coordinates of each point of the quadrilateral MNPQ will become:
\nM (1 + 2, 2 – 5), N (2 + 2, 4 – 5), P (4 + 2, 5 – 5), and Q (3 + 2, 3 – 5)
\nM (3, -3), N (4, -1), P (6, 0), and Q (5, -2)<\/p>\n

Question 4.
\nWhat are the coordinates of each point after quadrilateral MNPQ is reflected across the x-axis and then translated into 3 units left? Lessons 6-2 and 6-4
\nAnswer:
\nWe know that,
\nThe vertices of the quadrilateral MNPQ are:
\nM (1, 2), N (2, 4), P (4, 5), and Q (3, 3)
\nSo,
\nAfter the vertices of the quadrilateral, MNPQ reflected across the x-axis,
\nThe vertices of the quadrilateral MNPQ will become:
\nM (1, -2), N (2, -4), P (4, -5), and Q (3, -3)
\nNow,
\nAfter the Translation of 3 units left,
\nThe vertices of the quadrilateral MNPQ will become:
\nM (1 – 3, -2), N (2 – 3, -4), P (4 – 3, -5), and Q (3 – 3, -3)
\nM (-2, -2), N (-1, -4), P (1, -5), and Q (0, -3)
\nHence, from the above,
\nWe can conclude that the coordinates of each point after quadrilateral MNPQ is reflected across the x-axis and then translated into 3 units left are:
\nM (-2, -2), N (-1, -4), P (1, -5), and Q (0, -3)<\/p>\n

Question 5.
\nWhich series of transformations maps quadrilateral MNPQ onto quadrilateral RSTU? Lesson 6-4
\nA. reflection across the x-axis, translation 4 units down
\nB. reflection across the y-axis, translation 4 units down
\nC. rotation 180\u00b0 about the origin, and then reflection across the x-axis
\nD. rotation 180\u00b0 about the origin, and then reflection across the y-axis
\nAnswer:
\nThe representation\u00a0of the series of transformations that maps the quadrilateral MNPQ onto the quadrilateral RSTU is:
\n\"\"
\nHence, from the above,
\nWe can conclude that option A matches the given situation<\/p>\n

Question 6.
\nIs quadrilateral MNPQ congruent to quadrilateral RSTU? Explain. Lesson 6-5
\nAnswer:
\nWe know that,
\nThe 2 figures are said to be congruent only when:
\na. The shapes of the 2 figures are the same
\nb. The sizes of the 2 figures are the same (Side lengths, and Angle measures)
\nNow,
\n\"\"
\nFrom the above,
\nWe can observe that the side lengths are not the same
\nHence, from the above,
\nWe can conclude that the quadrilateral MNPQ is not congruent to the quadrilateral RSTU<\/p>\n

Topic 6 MID-TOPIC PERFORMANCE TASK<\/h3>\n

A tessellation is a design in a plane that uses one or more congruent figures, with no overlaps and no gaps, to cover the entire plane. A tessellation of an equilateral triangle is shown.
\n\"Envision<\/p>\n

PART A
\nExplain how the tessellation of an equilateral triangle is formed using reflections.
\nAnswer:
\nWe know that,
\nWhen you cut a shape out of paper, then flip it over, the flipped shape looks like a mirror image of the original shape. So a tessellation made with this technique is called a “Reflection tessellation”
\nNow,
\nSee if the figure will fit together with no gaps. The answer is yes, the figures will tessellate because it is made up of two shapes that do tessellate<\/p>\n

PART B
\nExplain how the tessellation of an equilateral triangle is formed using rotations.
\nAnswer:
\nWe know that
\nA “Rotational tessellation” is a pattern where the repeating shapes fit together by rotating 90 degrees
\nNow,
\nSee if the figure will fit together with no gaps. The answer is yes, the figures will tessellate because it is made up of two shapes that do tessellate<\/p>\n

PART C
\nWhich of the regular polygon(s) below can be tessellated using a series of transformations?
\n\"Envision
\nAnswer:
\nWe know that,
\nEquilateral triangles, squares, and regular hexagons\u00a0are the only regular polygons that will tessellate. Therefore, there are only three regular tessellations.
\nHence, from the above,
\nWe can conclude that square and pentagon can be tessellated by using a series of transformations<\/p>\n

Lesson 6.6 Describe Dilations<\/h3>\n

Solve & Discuss It!<\/strong>
\nA landscape architect designs a small splash pad represented by \u25b3ABC. Then she decides to make the splash pad larger as shown by \u25b3ADE. How are the splash pad designs alike? How are they different?
\n\"Envision
\nI can… dilate two-dimensional figures
\nAnswer:
\nIt is given that
\nA landscape architect designs a small splash pad represented by \u25b3ABC. Then she decides to make the splash pad larger as shown by \u25b3ADE.
\nNow,
\nThe given landscape architect is:
\n\"Envision
\nNow,
\nFrom the given landscape architect,
\nWe can observe that the difference in the splash designs
\nHence, from the above landscape architect,
\nWe can conclude that
\nThe splash designs are alike in:
\na. Shape b. Angle measures c. Orientation
\nThe splash designs are different in:
\na. Side lengths<\/p>\n

Look for Relationships
\nHow can you use what you know about scale drawings to compare and contrast the designs?
\nAnswer:
\nA dilation is a transformation that results in an image with the same shape, angle measures, and orientation as the preimage, but different side lengths.
\nWe know that,
\nWhen the scale factor is greater than 1, the dilation is a reduction.
\nWhen the scale factor is between 0 and 1, the dilation is an enlargement.<\/p>\n

Focus on math practices
\nReasoning Paul wants to make two square picnic tables. One table will have side lengths that are \\(\\frac{1}{2}\\) of the lengths of the second table. How do the tablets compare? Explain.
\nAnswer:
\nIt is given that
\nPaul wants to make two square picnic tables. One table will have side lengths that are \\(\\frac{1}{2}\\) of the lengths of the second table
\nSo,
\nThe size of the tables are in the ratio of \\(\\frac{1}{2}\\) : 1
\nSo,
\nThe size of the tables are in the ratio of 1:2
\nHence, from the above,
\nWe can conclude that the side lengths of one table are 2 times the side lengths of the second table<\/p>\n

Essential Question<\/strong>
\nWhat is the relationship between a preimage and its image after a dilation?
\nAnswer:
\nAfter dilation, the pre-image and image have the same shape but not the same size.
\nIn terms of Sides:
\nIn dilation, the sides of the pre-image and the corresponding sides of the image are proportional.<\/p>\n

Try It!<\/strong><\/p>\n

F’G’ H’I’ is the image of FGHI after a dilation with center at the origin. What is the scale factor?
\n\"Envision
\nThe ratio of a side length in FGHI to a corresponding side length in F’GH’I’is: \\(\\frac{}{}\\)
\nThe scale factor is __________.
\nAnswer:
\nIt is given that
\nF’G’ H’I’ is the image of FGHI after a dilation with center at the origin.
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of FGHI are:
\nF (1, 1), G (1, 2), H (2, 2), and I (2, 1)
\nThe vertices of F’G’H’I’ are:
\nF’ (5, 5), G’ (5, 10), H’ (10, 10), and I’ (10, 5)
\nWe know that,
\nThe “Scale factor” is the ratio of a length in the image to the corresponding length in the preimage
\nSo,
\nThe ratio of a side length in FGHI to a corresponding side length in F’GH’I’is:
\n\\(\\frac{Side length of F’G’}{Side length of FG}\\) (or) \\(\\frac{Side length of H’I’}{Side length of HI}\\)
\n= \\(\\frac{0 + 5}{0 + 1}\\) (or) \\(\\frac{0 + 5}{0 + 1}\\)
\n= 5
\nSo,
\nThe scale factor is: 5
\nHence, from the above,
\nWe can conclude that
\nThe ratio of a side length in FGHI to a corresponding side length in F’GH’I’is: \\(\\frac{5}{1}\\)
\nThe scale factor is 5.<\/p>\n

Convince Me!
\nQuadrilateral WXYZ is the image of quadrilateral FGHI after a dilation with center at the origin and a scale factor of 3.5. What are the coordinates of the vertices of quadrilateral WXYZ?
\nAnswer:
\nIt is given that
\nQuadrilateral WXYZ is the image of quadrilateral FGHI after a dilation with center at the origin and a scale factor of 3.5
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of FGHI are:
\nF (1, 1), G (1, 2), H (2, 2), and I (2, 1)
\nIt is given that
\nThe scale factor is: 3.5
\nSo,
\nThe vertices of the quadrilateral WXYZ are:
\nW ( 1 \u00d7 3.5, 1 \u00d73.5), X (1 \u00d7 3.5, 2 \u00d7 3.5), Y (2 \u00d7 3.5, 2 \u00d7 3.5), and Z (2 \u00d7 3.5, 1 \u00d7 3.5)
\nSo,
\nW (3.5, 3.5), X (3.5, 7), Y (7, 7), and Z (7, 3.5)
\nHence, from the above,
\nWe can conclude that
\nThe coordinates of the vertices of quadrilateral WXYZ are:
\nW (3.5, 3.5), X (3.5, 7), Y (7, 7), and Z (7, 3.5)<\/p>\n

Try It!<\/strong><\/p>\n

A dilation maps point L(3, 6) to its image L’ (2, 4). Complete the dilation of figure LMN and label the image L’M’N’. What is the scale factor? What is the length of side M’N’?
\n\"Envision
\nAnswer:
\nIt is given that
\nA dilation maps point L(3, 6) to its image L’ (2, 4). Complete the dilation of figure LMN and label the image L’M’N’
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the coordinate plane,
\nThe vertices of \u0394LMN are:
\nL (3, 6), M (3, 3), and N (6, 3)
\nIt is given that the image of L is: L’ (2, 4)
\nNow,
\nWe know that,
\nThe “Scale factor” is the ratio of a length in the image to the corresponding length in the preimage
\nSo,
\nThe scale factor = \\(\\frac{Side length of L’}{Side length of L}\\)
\n= \\(\\frac{4 – 2}{6 – 3}\\)
\n= \\(\\frac{2}{3}\\)
\nSo,
\nThe scale factor is: \\(\\frac{2}{3}\\)
\nNow,
\nThe coordinates of M’ = The coordinates of M \u00d7 \\(\\frac{2}{3}\\)
\n= (3, 3) \u00d7 \\(\\frac{2}{3}\\)
\n= (3 \u00d7 \\(\\frac{2}{3}\\), 3 \u00d7 \\(\\frac{2}{3}\\))
\n= (2, 2)
\nThe coordinates of N’ = The coordinates of N \u00d7 \\(\\frac{2}{3}\\)
\n= (6, 3) \u00d7 \\(\\frac{2}{3}\\)
\n= (6 \u00d7 \\(\\frac{2}{3}\\), 3 \u00d7 \\(\\frac{2}{3}\\))
\n= (4, 2)
\nNow,
\nCompare M’ and N’ with (x 1<\/sub>, y 1<\/sub>), (x 2<\/sub>, y 2 <\/sub>)
\nWe know that,
\nThe distance between 2 points = \u221a(x 2 <\/sub> – x 1 <\/sub>) 2 <\/sup> + (y 2 <\/sub> – y 1 <\/sub>) 2
\n<\/sup>So,
\nThe side length of M’N’ = \u221a(4\u00a0<\/sub>– 2\u00a0<\/sub>) 2 <\/sup> + (2\u00a0<\/sub>– 2\u00a0<\/sub>)2\u00a0<\/sup>
\n= 2 units
\nHence, from the above,
\nWe can conclude that
\nThe scale factor is: \\(\\frac{2}{3}\\)
\nThe side length of M’N’ is: 2 units<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

A dilation is a transformation that results in an image with the same shape, angle measures, and orientation as the preimage, but different side lengths.
\nWhen the scale factor is greater than 1, the dilation is an enlargement.
\n\"Envision<\/p>\n

When the scale factor is between 0 and 1, the dilation is a reduction.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> What is the relationship between a preimage and its image after a dilation?
\nAnswer:
\nAfter dilation, the pre-image and image have the same shape but not the same size.
\nIn terms of Sides:
\nIn dilation, the sides of the pre-image and the corresponding sides of the image are proportional.<\/p>\n

Question 2.
\nGeneralize When will a dilation be a reduction? When will it be an enlargement?
\nAnswer:
\nWhen the scale factor is greater than 1, the dilation is an enlargement.
\n\"Envision
\nWhen the scale factor is between 0 and 1, the dilation is a reduction.
\n\"Envision<\/p>\n

Question 3.
\nReasoning<\/strong> Flora draws a rectangle with points at (12, 12), (15, 12), (15,9), and (12, 9). She dilates the figure with center at the origin and a scale factor of \\(\\frac{3}{4}\\). what is the measure of each angle in the image? Explain.
\nAnswer:
\nIt is given that
\nFlora draws a rectangle with points at (12, 12), (15, 12), (15,9), and (12, 9). She dilates the figure with center at the origin and a scale factor of \\(\\frac{3}{4}\\)
\nSo,
\nThe vertices for the image of the given rectangle = (x \u00d7 \\(\\frac{3}{4}\\), y \u00d7 \\(\\frac{3}{4}\\))
\nSo,
\nThe vertices for the image of the given rectangle wil be:
\n( 12 \u00d7 \\(\\frac{3}{4}\\), 12 \u00d7 \\(\\frac{3}{4}\\)), (15 \u00d7 \\(\\frac{3}{4}\\), 12 \u00d7 \\(\\frac{3}{4}\\)), (15 \u00d7 \\(\\frac{3}{4}\\), 9 \u00d7 \\(\\frac{3}{4}\\)), and (12 \u00d7 \\(\\frac{3}{4}\\), 9 \u00d7 \\(\\frac{3}{4}\\))
\n= (9, 9), (11.25, 9), (11.25, 6.75), and (9, 6.75)
\nNow,
\nWe know that,
\nIn dilation,
\na. The preimage and image are the same in shape, orientation, and angle measures
\nb. the preimage and the image are different in size and the side lengths
\nSo,
\nThe representation of the angle measures in the preimage and image of the given rectangle is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the measure of each angle in the image is: 90\u00b0<\/p>\n

Do You Know How?<\/strong>
\nIn 4-6, use the coordinate grid below.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the coordinate plane,
\nThe vertices of Figure 1 are:
\n(4, 4), (6, 8), and (8, 4)
\nThe vertices of Figure 2 are:
\n(2, 2), (3, 4), and (4, 2)
\nThe vertices of Figure 3 are:
\n(1, 1), (1.5, 2), and (2, 1)<\/p>\n

Question 4.
\nFigure 3 is the image of Figure 1 after a dilation with a center at the origin. What is the scale factor? Explain.
\nAnswer:
\nIt is given that
\nFigure 3 is the image of Figure 1 after a dilation with a center at the origin
\nNow,
\nWe know that,
\nThe “Scale factor” is the ratio of a length in the image to the corresponding length in the preimage
\nSo,
\nScale factor = \\(\\frac{4}{1}\\) (or) \\(\\frac{6}{1.5}\\) (or) \\(\\frac{8}{2}\\)
\n= 4
\nHence, from the above,
\nWe can conclude that the scale factor is “4” so that Figure 3 is the image of Figure 1<\/p>\n

Question 5.
\nWhat are the coordinates of the image of Figure 2 after a dilation with center at the origin and a scale factor of 3?
\nAnswer:
\nWe know that,
\nThe vertices of Figure 2 are:
\n(2, 2), (3, 4), and (4, 2)
\nSo,
\nWith a scale factor of 3,
\nThe vertices of Figure 2 will become:
\n(2 \u00d7 3, 2 \u00d7 3), (3 \u00d7 3, 4 \u00d7 3), and (4 \u00d7 3, 2 \u00d7 3)
\n= (6, 6), (9, 12), and (12, 6)
\nHence, from the above,
\nWe can conclude that the coordinates of the image of Figure 2 after a dilation at the origin and a scale factor of 3 are:
\n(6, 6), (9, 12), and (12, 6)<\/p>\n

Question 6.
\nWhich figures represent a dilation with a scale factor of \\(\\frac{1}{2}\\)?
\nAnswer:
\nWe know that,
\nThe vertices of Figure 1 are:
\n(4, 4), (6, 8), and (8, 4)
\nThe vertices of Figure 2 are:
\n(2, 2), (3, 4), and (4, 2)
\nThe vertices of Figure 3 are:
\n(1, 1), (1.5, 2), and (2, 1)
\nNow,
\nIf we consider Figure 2 as the preimage and Figure 1 as the image, then
\nScale factor = \\(\\frac{2}{4}\\) (or) \\(\\frac{3}{6}\\) (or) \\(\\frac{4}{8}\\)
\n= \\(\\frac{1}{2}\\)
\nIf we consider Figure 3 as the preimage and Figure 2 as the image, then
\nScale factor = \\(\\frac{1}{2}\\) (or) \\(\\frac{1.5}{3}\\) (or) \\(\\frac{2}{4}\\)
\n= \\(\\frac{1}{2}\\)
\nHence, from the above,
\nWe can conclude that (Figure 2, Figure 1) and (Figure 3, Figure 2) represent a dilation with a scale factor of \\(\\frac{1}{2}\\)<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nLeveled Practice Draw the image of \u25b3DEF after a dilation with center (0, 0) and scale factor of 2.
\n\"Envision
\nFind the coordinates of each point in the original figure.
\nD(____, (____) E(____, (____) F(____, (____)
\nMultiply each coordinate by _______.
\nFind the coordinates of each point in the image:
\nD'(____, (____) E'(____, (____) F'(____, (____)
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of \u0394DEF are:
\nD (0, 0), E (2, 0), and F (0, 2)
\nNow,
\nWith a scale factor of 2,
\nMultiply each coordinate with 2
\nThe coordinates of \u0394DEF will become:
\nD (0 \u00d7 2, 0 \u00d7 2), E (2 \u00d7 2, 0 \u00d7 2), and F (0 \u00d7 2, 2 \u00d7 2)
\nD (0, 0), E (4, 0), and F (0, 4)
\nSo,
\nThe coordinates of the points for the image of \u0394DEF are:
\nD’ (0, 0), E’ (4, 0), and F’ (0, 4)
\nHence,
\nThe representation of \u0394DEF and its image \u0394D’E’F’ is:
\n\"\"<\/p>\n

Question 8.
\nFind the scale factor for the dilation shown.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of Figure DEFG are:
\nD (0, 0), E (5, 0), F (5, 6), and G (0, 6)
\nThe vertices of Figure D’E’F’G’ are:
\nD’ (0, 0), E’ (15, 0), G’ (15, 18), and H’ (0, 18)
\nNow,
\nWe know that,
\nThe “Scale factor” is the ratio of a length in the image to the corresponding length in the preimage
\nSo,
\nScale factor = \\(\\frac{15}{5}\\) (or) \\(\\frac{18}{6}\\)
\n= 3
\nHence, from the above,
\nWe can conclude that
\nThe scale factor for the given dilation is: 3<\/p>\n

Question 9.
\nCritique Reasoning For the dilation with center (0, 0) shown on the graph, your friend says the scale factor is \\(\\frac{5}{2}\\). What is the correct scale factor? What mistake did your friend likely make?
\n\"Envision
\nAnswer:
\nIt is given that
\nFor the dilation with center (0, 0) shown on the graph, your friend says the scale factor is \\(\\frac{5}{2}\\)
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394ABC are:
\nA (0, 5), B (-5, 0), and C (5, -5)
\nThe vertices of \u0394A’B’C’ are:
\nA’ (0, 2), B’ (-2, 0), C’ (2, -2)
\nNow,
\nWe know that,
\nThe “Scale factor” is the ratio of a length in the image to the corresponding length in the preimage
\nSo,
\nScale factor = \\(\\frac{2}{5}\\) (or) \\(\\frac{-2}{-5}\\)
\n= \\(\\frac{2}{5}\\)
\nHence, from the above,
\nWe can conclude that
\nThe correct scale factor is: \\(\\frac{2}{5}\\)
\nThe mistake done by your friend is:
\nThe consideration of the scale factor as \\(\\frac{Length of the preimage}{length of the image}\\) instead of \\(\\frac{Length of the image}{Length of the preimage}\\)<\/p>\n

Question 10.
\nThe smaller figure is the image of dilation of the larger figure. The origin is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe smaller figure is the image of dilation of the larger figure. The origin is the center of dilation
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe coordinates of each point of the preimage are:
\n(3, 6), (15, 6), (15, 18), and (3, 18)
\nThe coordinates of each point of the image are:
\n(1, 2), (5, 2), (5, 6), and (1, 6)
\nNow,
\nWe know that,
\nThe “Scale factor” is the ratio of a length in the image to the corresponding length in the preimage
\nSo,
\nScale factor = \\(\\frac{1}{3}\\) (or) \\(\\frac{2}{6}\\) (or) \\(\\frac{5}{15}\\) (or) \\(\\frac{6}{18}\\)
\n= \\(\\frac{1}{3}\\)
\nNow,
\nSince the scale factor is less than 1, the dilation is a reduction
\nHence, from the above,
\nWe can conclude that
\nThe given dilation is a reduction
\nThe scale factor is: \\(\\frac{1}{3}\\)<\/p>\n

Question 11.
\nHigher-Order Thinking<\/strong> Q’R’S’T’ is the image of QRST after a dilation with center at the origin.
\n\"Envision
\nAnswer:
\nIt is given that
\nQ’R’S’T’ is the image of QRST after a dilation with center at the origin
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of the parallelogram QRST are:
\nQ (4, 4), R (16, 4), S (20, 16), and T (8, 16)
\nThe vertices of the parallelogram Q’R’S’T’ are:
\nQ’ (1, 1), R’ (4, 1), S’ (5, 4), and T’ (2, 4)<\/p>\n

a. Find the scale factor.
\nAnswer:
\nWe know that,
\nThe “Scale factor” is the ratio of a length in the image to the corresponding length in the preimage
\nSo,
\nScale factor = \\(\\frac{4}{1}\\) (or) \\(\\frac{16}{4}\\) (or) \\(\\frac{20}{5}\\) (or) \\(\\frac{8}{2}\\)
\n= 4
\nHence, from the above,
\nWe can conclude that the scale factor is: 4<\/p>\n

b. Find the area of each parallelogram. What is the relationship between the areas?
\nAnswer:
\nWe know that,
\nArea of the parallelogram = Base \u00d7 Height
\nNow,
\nthe representation of the side lengths of the parallelogram QRST and its dilation Q’R’S’T’ is:
\n\"\"
\nSo,
\nThe area of the parallelogram QRST = 12.6 \u00d7 12
\n= 151.2 square units
\nThe area of the parallelogram Q’R’S’T’ = \\(\\frac{The area of the parallelogram QRST}{4}\\)
\n= 37.8 square units
\nHence, from the above,
\nWe can conclude that
\nThe area of the parallelogram QRST is: 151.2 sq. units
\nThe area of the parallelogram Q’R’S’T’ is: 37.8 sq. units
\nThe relationship between the two areas is:
\nThe area of the parallelogram Q’R’S’T’ = \\(\\frac{The area of the parallelogram QRST}{4}\\)<\/p>\n

Assessment Practice<\/strong>
\nQuestion 12.
\nTriangle PQR is the image of \u25b3JKL after a dilation. Is the dilation an enlargement or a reduction? Explain.
\n\"Envision
\nA. An enlargement, because the image is larger than the original figure
\nB. An enlargement, because the image is smaller than the original figure
\nC. A reduction, because the image is smaller than the original figure
\nD. A reduction, because the image is larger than the original figure
\nAnswer:
\nIt is given that
\nTriangle PQR is the image of \u25b3JKL after a dilation
\nNow,
\nFrom the given figure,
\nWe can observe that \u0394PQR is larger than \u0394JKL
\nSo,
\nWe can say that the dilation is an enlargement
\nHence, from the above,
\nWe can conclude that option A matches with the given situation<\/p>\n

Question 13.
\nRectangle QUAD has coordinates Q(0, 0), U(0, 3), A(6, 3), and D(6, 0). Q’U’ A’D’ is the image of QUAD after a dilation with center (0, 0) and a scale factor of 6. What are the coordinates of point D’? Explain.
\nAnswer:
\nIt is given that
\nRectangle QUAD has coordinates Q(0, 0), 4(0, 3), A6, 3), and D(6, 0). Q’U’ A’D’ is the image of QUAD after a dilation with center (0, 0) and a scale factor of 6.
\nSo,
\nWith a scale factor of 6,
\nThe vertices of the rectangle QUAD will become:
\nQ (0 \u00d7 6, 0 \u00d7 6), U (0 \u00d7 6, 3 \u00d7 6), A (6 \u00d7 6, 3 \u00d7 6) and D (6 \u00d7 6, 0 \u00d7 6)
\n= Q (0, 0), U (0, 18), A (36, 18), and D (6, 0)
\nSo,
\nAfter a dilation,
\nThe vertices of the rectangle QUAD will become the vertices of the rectangle Q’U’A’D’
\nSo,
\nThe vertices of the rectangle Q’U’A’D’ are:
\nQ’ (0, 0), U’ (0, 18), A’ (36, 18), and D’ (36, 0)
\nHence, from the above,
\nWe can conclude that the coordinates of point D’ are: (36, 0)<\/p>\n

Lesson 6.7 Understand Similar Figures<\/h3>\n

Solve & Discuss It!<\/strong>
\nAndrew draws the two figures shown on a coordinate plane. How are the two figures alike? How are they different? How do you know?
\n\"Envision
\nI can… use a sequence of transformations, including dilations, to show that figures are similar.
\nAnswer:
\nIt is given that
\nAndrew draws the two figures shown on a coordinate plane
\nNow,
\nThe given figures are:
\n\"Envision
\nFrom the above,
\nWe can observe that
\n\u0394ABC is the preimage
\n\u0394A’B’C’ is the image
\nWe can also observe that
\nThe image is smaller than the image since the transformation we used when drawing the image is a “Dilation”
\nHence,
\nSince the transformation we used is a “Dilation”,
\nThe two figures are similar in terms of:
\na. Shape b. Size c. Orientation d. Angle measures
\nThe two figures are different in terms of:
\na. Side lengths<\/p>\n

Look for Relationships
\nIs \u25b3ABC a preimage of \u25b3A’B’C’? How do you know?
\nAnswer:
\nThe given figures are:
\n\"Envision
\nFrom the given figures,
\nWe can clearly observe that the transformation called “Dilation” takes place
\nSince the dilation takes place,
\nWe can conclude that \u0394ABC is a preimage of \u0394A’B’C’<\/p>\n

Focus on math practices
\nReasoning How can you use the coordinates of the vertices of the triangles to identify the transformation that maps \u25b3ABC to \u25b3A’B’C’? Explain.
\nAnswer:
\nThe given figures are:
\n\"Envision
\nNow,
\nFro the given figures,
\nWe know that a transformation called a “Dilation” takes place
\nSo,
\nWe know that,
\nThe “Scale factor” is the ratio of a length in the image to the corresponding length in the preimage
\nSo,
\nScale factor = [altex]\\frac{The x-coordinate (or) y-coordinate that maps A’ or B’ or C’}{The x-coordinate (or) y-coordinate that maps A or B or C}[\/latex}
\nHence, from the above,
\nWe can conclude that
\nBy using the “Scale factor”, the coordinates of the vertices of the triangles can be used to identify the transformation that maps \u25b3ABC to \u25b3A’B’C’<\/p>\n

Essential Question<\/strong>
\nHow are similar figures related by a sequence of transformations?
\nAnswer:
\nTwo figures are similar if and only if one figure can be obtained from the other by a single transformation, or a sequence of transformations, including translations, reflections, rotations, and\/or dilations. Similarity transformations preserve shape, but not necessarily size, making the figures “similar”<\/p>\n

Try It!<\/strong><\/p>\n

Is \u25b3\u0391\u0392C similar to \u25b3A’ B’ C?
\nThe triangles _________ similar.
\n\"Envision
\nAnswer:
\nThe given figures are:
\n\"Envision
\nNow,
\nIf we want to find the 2 triangles are similar or not, find the scale factor and find whether they have the same shape or not
\nIf the scale factor for 2 triangles is the same, then those triangles are similar
\nNow,
\nWe know that,
\nThe scale factor is defined as the distance from this center point to a point on the preimage and also the distance from the center point to a point on the image
\nNow,
\nFor AB and A’B’,
\nScale factor = \\(\\frac{27}{12}\\)
\n= \\(\\frac{9}{4}\\)
\nFor BC and B’C’,
\nScale factor = \\(\\frac{27}{12}\\)
\n= \\(\\frac{9}{4}\\)
\nFor AC and A’C’,
\nScale factor = \\(\\frac{18}{8}\\)
\n= \\(\\frac{9}{4}\\)
\nSince the scale factor is equal for all the corresponding sides
\nWe can conclude that \u0394ABC is similar to \u0394A’B’C’<\/p>\n

Convince Me!
\nWhat sequence of transformations shows that \u25b3ABC is similar to \u25b3A’ B’C’?
\nAnswer:
\nThe sequence of Transformations that shows \u0394ABC is similar to \u0394A’B’C’ is:
\na. Rotation b. Dilation c. Translation<\/p>\n

Try It!<\/strong><\/p>\n

a. Graph the image of JKL after a reflection across the line x = 1 followed by dilation with a scale factor of \\(\\frac{1}{2}\\) and center of dilation point J’.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394JKL are:
\nJ (-2, -4), K (-4, 0), and L (-2, 1)
\nThe vertices of \u0394PQR are:
\nP (-4, 2), Q (4, 6), and R (6, 2)
\nNow,
\nAfter the reflection of \u0394JKL across the line x = 1,
\nJ (-1, 4), K (-3, 0), and L (-1, -1)
\nSo,
\nWith a scale factor of \\(\\frac{1}{2}\\),
\nThe vertices of \u0394JKL will become:
\nJ (-1, 4) \u00d7 \\(\\frac{1}{2}\\), K (-3, 0) \u00d7 \\(\\frac{1}{2}\\), and L (-1, -1) \u00d7 \\(\\frac{1}{2}\\)
\nSo,
\nJ’ (-0.5, 2), K’ (-1.5, 0), and L’ (-0.5, -0.5)
\nHence,
\nThe representation of the image of JKL after a reflection across the line x = 1 followed by dilation with a scale factor of \\(\\frac{1}{2}\\) and center of dilation point J’ is:
\n\"\"<\/p>\n

b. Is \u25b3JKL similar to \u25b3PQR?
\nAnswer:
\nWe know that,
\nThe vertices of \u0394JKL are:
\nJ (-2, -4), K (-4, 0), and L (-2, 1)
\nThe vertices of \u0394PQR are:
\nP (-4, 2), Q (4, 6), and R (6, 2)
\nNow,
\nStep 1:
\nRotate the vertices of \u0394JKL 270\u00b0 about the origin
\nWe know that,
\n(x, y) before 270\u00b0 rotation —-> (y, -x) after 270\u00b0 rotation
\nSo,
\nThe vertices of \u0394JKL will be:
\nJ (-4, 2), K (0, 4), and L (1, 2)
\nStep 2:
\nDilate the vertices of \u0394JKL we obtained in step 1 by a scale factor of 2
\nSo,
\nThe vertices of \u0394JKL will be:
\nJ (-8, 4), K (0, 8), and L (2, 4)
\nStep 3:
\nTranslate the vertices of \u0394JKL we obtained in step 2 by 4 units right and 2 units down
\nSo,
\nThe vertices of \u0394JKL will become:
\nP (-4, 2), Q (4, 6), and R (6, 2)
\nSo,
\nThe representation of the sequence of transformations is:
\n\"\"
\nHence, from the above,
\nWe can conclude that \u0394JKL is similar to \u0394PQR<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

Two-dimensional figures are similar if there is a sequence of rotations, reflections, translations, and dilations that maps one figure onto the other.<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How are similar figures related by a sequence of transformations?
\nAnswer:
\nTwo figures are similar if and only if one figure can be obtained from the other by a single transformation, or a sequence of transformations, including translations, reflections, rotations, and\/or dilations. Similarity transformations preserve the shape, but not necessarily size, making the figures “similar”<\/p>\n

Question 2.
\nBe Precise How do the angle measures and side lengths compare in similar figures?
\nAnswer:
\nMatching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles. If two figures are similar, then the\u00a0measures of the corresponding angles are equal and the ratios of the lengths of the corresponding sides are proportional.<\/p>\n

Question 3.
\nGeneralize Does a given translation, reflection, or rotation, followed by a given dilation, always map a figure to the same image as that same dilation followed by that same translation, reflection, or rotation? Explain.
\nAnswer:
\nWe know that,
\nThe images formed by reflection, translation, and rotation have the same shape and size whereas dilation has the same shape but a different size
\nIn the same way,
\nThe image formed by dilation has the same shape but a different size whereas the image formed by reflection, translation, and rotation has the same shape and size as the image formed in dilation
\nHence, from the above,
\nWe can conclude that
\nThe translation, reflection, or rotation, followed by a given dilation, always map a figure to the same image as that same dilation followed by that same translation, reflection, or rotation<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nIs trapezoid ABCD – trapezoid EFGH? Explain.
\n\"Envision
\nAnswer:
\nThe given figures are:
\n\"Envision
\nFrom the given figures,
\nWe can observe that
\nThe corresponding angles i.e., the opposite angles are equal
\nNow,
\nThe ratio of the side lengths of the corresponding figures = \\(\\frac{32}{16}\\) (or) \\(\\frac{28}{14}\\) (or) \\(\\frac{20}{10}\\)
\n= 2
\nSo,
\nThe ratio of the side lengths of the corresponding figures are the same
\nNow,
\nWe know that,
\nTwo figures are said to be similar if they are the same shape. In more mathematical language, two figures are similar if their corresponding angles are congruent, and the ratios of the lengths of their corresponding sides are equal. This common ratio is called the scale factor
\nHence, from the above,
\nWe can conclude that
\nTrapezoid ABCD – Trapezoid EFGH<\/p>\n

Use the graph for 5 and 6.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394ABC are:
\nA (-4, -2), B (-1, -2), and C (-3, -3)
\nThe vertices of \u0394DEF are:
\nD (9, -5), E (6, -8), and F (1, -5)<\/p>\n

Question 5.
\n\u25b3ABC is dilated by a factor of 2 with a center of dilation at point C, reflected across the x-axis, and translated 3 units up. Graph the resulting similar figure.
\nAnswer:
\nWe know that,
\nThe vertices of \u0394ABC are:
\nA (-4, -2), B (-1, -2), and C (-3, -3)
\nSo,
\nStep 1:
\nAfter dilation by a factor of 2:
\nMultiply all the coordinates of each point by 2
\nSo,
\nThe vertices of \u0394ABC are:
\nA (-8, -4), B (-2, -4), and C (-6, -6)
\nNow,
\nWith a dilation at point C,
\nAdd all the coordinates of each point with -3
\nSo,
\nThe vertices of \u0394ABC are:
\nA (-11, -7), B (-5, -7), and C (-9, -9)
\nStep 2:
\nWhen the vertices of \u0394ABC that we obtained in step 1 reflects across the x-axis,
\nThe vertices of \u0394ABC will be:
\nA (-11, 7), B (-5, 7), and C (-9, 9)
\nStep 3:
\nWhen the vertices of \u0394ABC we obtained in step 2 translated by 3 units up,
\nAdd all the y-coordinates of each point with 3
\nSo,
\nThe vertices of \u0394ABC will be:
\nA (-11, 10), B (-5, 10), and C (-9, 12)
\nHence,
\nThe representation of all the steps is:
\n\"\"<\/p>\n

Question 6.
\nIs \u25b3ABC similar to \u25b3DEF? Explain.
\nAnswer:
\nWe know that,
\nThe vertices of \u0394ABC are:
\nA (-4, -2), B (-1, -2), and C (-3, -3)
\nThe vertices of \u0394DEF are:
\nD (9, -5), E (6, -8), and F (1, -5)
\nWe know that,
\nTwo figures are said to be similar if they are the same shape. In more mathematical language, two figures are similar if their corresponding angles are congruent, and the ratios of the lengths of their corresponding sides are equal
\nSo,
\nThe representation of \u0394ABC and \u0394DEF are:
\n\"\"
\nFrom the above,
\nWe can observe that the ratio of the corresponding side lengths are not in the same proportion
\nHence, from the above,
\nWe can conclude that \u0394ABC is not similar to \u0394DEF<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nLeveled Practice RSTU and VXYZ are quadrilaterals. Given RSTU ~ VXYZ, describe a sequence of transformations that maps RSTU to VXYZ.
\n\"Envision
\n\u2022 reflection across the ________
\n\u2022 translation ______ unit(s) left and _______ unit(s) up
\n\u2022 dilation with center (0,0) and a scale factor of ________
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the coordinate plane,
\nThe vertices of the quadrilateral VXYZ are:
\nV (-2, -1), X (-3, 0), Y (-2, 2), and Z (-1, 0)
\nThe vertices of the quadrilateral RSTU are:
\nR (3, 0), S (6, 3), T (3, 9), and U (0, 3)
\nNow,
\nStep 1:
\nReflect the vertices of the quadrilateral VXYZ across the y-axis
\nSo,
\nThe vertices of the quadrilateral VXYZ will be:
\nV (2, -1), X (3, 0), Y (2, 2), and Z (1, 0)
\nStep 2:
\nDilate the vertices of the quadrilateral VXYZ with a scale factor of 3
\nSo,
\nMultiply all the coordinates of all the vertices of the quadrilateral VXYZ by 3
\nSo,
\nThe vertices of the quadrilateral VXYZ will be:
\nV (6, -3), X (9, 0), Y (6, 6), and Z (3, 0)
\nStep 3:
\nTranslate the vertices of the quadrilateral VXYZ 3 units left and 3 units up
\nSo,
\nThe vertices of the quadrilateral VXYZ will become:
\nR (3, 0), S (6, 3), T (3, 9), and U (0, 3)
\nHence,
\nThe sequence of Transformations that map the quadrilateral VXYZ onto the quadrilateral RSTU is:
\n\"\"<\/p>\n

Question 8.
\nReasoning Is \u25b3MNO similar to \u25b3PQO? Explain.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the coordinate plane,
\nThe vertices of \u0394MNO are:
\nM (6, 6 ), N (0, 6), and O (0, 0)
\nThe vertices of \u0394PQO are:
\nP (-12, -9), Q (0, -9), and O (0, 0)
\nWe know that,
\nTwo figures are said to be similar if they are the same shape. In more mathematical language, two figures are similar if their corresponding angles are congruent, and the ratios of the lengths of their corresponding sides are equal
\nSo,
\nThe representation of \u0394MNO and \u0394PQO are:
\n\"\"
\nFrom the above,
\nWe can observe that
\na. The ratio of the corresponding side lengths are not in the same proportion
\nb. The angle measures are not the same
\nHence, from the above,
\nWe can conclude that \u0394MNO is not similar to \u0394PQO<\/p>\n

Question 9.
\n\u25b3PQR is dilated by a scale factor of 2 with a center of dilation (0, 0) and rotated 180\u00b0 about the origin. Graph the resulting similar \u25b3XYZ.
\n\"Envision
\nAnswer:
\nIt is given that
\n\u25b3PQR is dilated by a scale factor of 2 with a center of dilation (0, 0) and rotated 180\u00b0 about the origin.
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394PQR are:
\nP (2, 2), Q (4, 2), and R (3, 4)
\nNow,
\nStep 1:
\nDilation of \u0394PQR by a scale factor of 2:
\nMultiply all the coordinates of each point of \u0394PQR by 2
\nSo,
\nThe vertices of \u0394PQR will be:
\nP (4, 4), Q (8, 4), and R (6, 8)
\nStep 2:
\nRotation of 180\u00b0 counterclockwise about the origin:
\nWe know that,
\n(x, y) before rotating 180\u00b0 ——- > (-x, -y) after rotating 180\u00b0
\nSo,
\nThe vertices of \u0394PQR that we obtained in step 1 will become:
\nX (-4, -4), Y (-8, -4), and Z (-6, -8)
\nHence,
\nThe graph of the resulting \u0394XYZ is:
\n\"\"<\/p>\n

Question 10.
\nDescribe a sequence of transformations that shows that quadrilateral RSTU is similar to quadrilateral VXYZ.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of the quadrilateral VXYZ are:
\nV (-5, -5), X (-5, -1), Y (-1, -1), and Z (-1, -5)
\nThe vertices of the quadrilateral RSTU are:
\nR (-16, -14), S (-16, -6), T (-8, -6), and U (-8, -14)
\nNow,
\nStep 1:
\nTranslate the quadrilateral VXYZ 3 units left and 2 units down
\nSo,
\nThe vertices of the quadrilateral VXYZ will be:
\nV (-8, -7), X (-8, -3), Y (-4, -3), and Z (-4, -7)
\nStep 2:
\nDilate the vertices of the quadrilateral VXYZ we obtained in step 1 by a scale factor of 2
\nSo,
\nThe vertices of the quadrilateral VXYZ will be:
\nR (-16, -14), S (-16, -6), T (-8, -6), and U (-8, -14)
\nHence,
\nThe sequence of Transformations that shows the quadrilateral VXYZ is similar to the quadrilateral RSTU is:
\n\"\"<\/p>\n

Question 11.
\nConstruct Arguments Is \u25b3PQR similar to \u25b3XYZ? Explain.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394XYZ are:
\nX (4, 4), Y (4, 8), and Z (8, 8)
\nThe vertices of \u0394PQR are:
\nP (2, -2), Q (4, -2), and R (4, -4)
\nNow,
\nStep 1:
\nRotate \u0394XYZ 270\u00b0 about the origin
\nWe know that,
\n(x, Y) before 270\u00b0 rotation —— > (y, -x) after 270\u00b0 rotation
\nSo,
\nThe vertices of \u0394XYZ will become:
\nX ( 4, -4), Y (8, -4), and Z (8, -8)
\nStep 2:
\nDilate \u0394XYZ with a scale factor of \\(\\frac{1}{2}\\)
\nSo,
\nThe vertices of \u0394XYZ that we obtained in step 1 will become:
\nP (2, -2), Q (4, -2), and R (4, -4)
\nSo,
\nAfter the sequence of Rotation and Dilation of \u0394XYZ,
\nWe obtained the vertices of \u0394PQR
\nHence, from the above,
\nWe can conclude that \u0394PQR is similar to \u0394XYZ<\/p>\n

Question 12.
\nHigher-Order Thinking<\/strong> Given \u25b3JKL ~ \u25b3XYZ, find two possible coordinates for missing point Y. For each coordinate chosen, describe a sequence of transformations, including a dilation, that will map \u25b3JKL to \u25b3XYZ.
\n\"Envision
\nAnswer:
\nIt is given that
\nGiven \u25b3JKL ~ \u25b3XYZ
\nNow,
\nThe given coordinate plane is:
\n\"Envision
\nFrom the coordinate plane,
\nThe vertices of \u0394JKL are:
\nJ (2, 8), K (6, 2), and L (2, 2)
\nThe vertices of \u0394XYZ are:
\nX (-2, 5), Y (x, y), and Z (-2, 2)
\nNow,
\nStep 1:
\nReflect \u0394JKL across the y-axis
\nSo,
\nThe vertices of \u0394JKL will be:
\nJ (-2, 8), K (-6, 2), and L (-2, 2)
\nStep 2:
\nDilate the vertices of \u0394JKL that we obtained in step by a scale factor of \\(\\frac{1}{2}\\)
\nSo,
\nThe vertices of \u0394JKL will be:
\n(-1, 4), K (-3, 1), and L (-1, 1)
\nStep 3:
\nTranslate the vertices of \u0394JKL that we obtained in step 2 by 1 unit left and 1 unit up
\nSo,
\nThe vertices of \u0394JKL will become:
\nX (-2, 5), Y (-4, 2), and Z (-2, 2)
\nHence, from the above,
\nWe can conclude that the coordinates for the missing point Y are: (-4, 2)<\/p>\n

Assessment Practice<\/strong>
\nQuestion 13.
\nRajesh is making pennants in preparation for a school soccer game. He wants the pennants to be similar triangles. Which of these triangles could he use for the pennants?
\n\"Envision
\nA. \u25b3QRS and \u25b3TVW
\nB. \u25b3QRS and \u25b3XYZ
\nC. \u25b3TVW and \u25b3JKL
\nD. \u25b3TVW and \u25b3XYZ
\nAnswer:
\nIt is given that
\nRajesh is making pennants in preparation for a school soccer game. He wants the pennants to be similar triangles.
\nNow,
\nIf we want to find which of the triangles are similar, find the scale factor and find whether they have the same shape or not
\nIf the scale factor for 2 triangles is the same, then those triangles are similar
\nNow,
\nWe know that,
\nThe scale factor is defined as the distance from this center point to a point on the preimage and also the distance from the center point to a point on the image
\nNow,
\nFor \u0394QRS and \u0394TVW,
\nScale factor = \\(\\frac{3}{1}\\) (or) \\(\\frac{3}{1.5}\\)
\n= 3 (or) 2
\nFor \u0394QRS and \u0394XYZ,
\nScale factor = \\(\\frac{3}{1.5}\\) (or) \\(\\frac{3}{1.5}\\)
\n= 2
\nHence from the above,
\nWe can conclude that \u0394QRS and \u0394XYZ are similar triangles and he can use these triangles for the pennants<\/p>\n

Question 14.
\nDetermine whether the following pairs of triangles are similar or not similar.
\n\"Envision
\nAnswer:
\nIf we want to find which of the triangles are similar, find the scale factor and find whether they have the same shape or not
\nIf the scale factor for 2 triangles is the same, then those triangles are similar
\nNow,
\nWe know that,
\nThe scale factor is defined as the distance from this center point to a point on the preimage and also the distance from the center point to a point on the image
\nNow,
\nFor \u0394ABC and \u0394DEF,
\nScale factor = \\(\\frac{2}{1}\\) (or) \\(\\frac{2}{1}\\)
\n= 2
\nFor \u0394ABC and \u0394LMN,
\nScale factor = \\(\\frac{2}{1.5}\\) (or) \\(\\frac{2}{1}\\)
\n= 1.33 (or) 2
\nFor \u0394DEF and \u0394LMN,
\nScale factor = \\(\\frac{1}{1}\\) (or) \\(\\frac{1.5}{1}\\)
\n= 1 (or) 1.5
\nHence from the above,
\nWe can conclude that \u0394ABC and \u0394DEF are similar triangles<\/p>\n

Lesson 6.8 Angles, Lines, and Transversals<\/h3>\n

Solve & Discuss It!<\/strong>
\nDraw two parallel lines. Then draw a line that intersects both lines. Which angles have equal measures?
\n\"Envision
\nI can… identify and find the measures of angles formed by parallel lines and a transversal.
\nAnswer:
\nLet a and b be the two parallel lines and the line that is intersecting the parallel lines is a “Transversal”
\nSo,
\nThe representation of the parallel lines along with the transversal is:
\n\"\"
\nNow,
\nFrom the above,
\nThe angles that have equal measures are:
\nCorresponding angles:
\n\u22201 = \u22205, \u22202 = \u22206, \u22203 = \u22207, and \u22204 = \u22208
\nAlternate Interior angles:
\n\u22204 = \u22206, \u22203 = \u22205
\nAlternate Exterior angles:
\n\u22201 = \u22207, \u22202 = \u22208
\nVertically Opposite Angles:
\n\u22201 = \u22207, \u22202 = \u22208<\/p>\n

Use Appropriate Tools
\nWhat tools can you use to determine which angles have equal measures?
\nAnswer:
\nThe tools that can be used to determine which angles have equal measures are:
\na. Pencil b. Scale c. Protractor d. Setsquare<\/p>\n

Focus on math practices
\nReasoning What properties or definitions can you use to describe which angles have equal measures?
\nAnswer:
\nThe definitions that you can use to describe which angles have equal measures are:
\nCorresponding angles:
\nCorresponding angles are\u00a0angles that are in the same position relative to lines intersected by a transversal
\nAlternate Interior angles:
\nWhen two parallel lines are crossed by a transversal, the pair of angles formed on the inner side of the parallel lines, but on the opposite sides of the transversal are called alternate interior angles
\nAlternate Exterior angles:
\nAlternate exterior angles are\u00a0the pair of angles that lie on the outer side of the two parallel lines but on either side of the transversal line<\/p>\n

Essential Question<\/strong>
\nWhat are the relationships among angles that are created when a line intersects two parallel lines?
\nAnswer:
\nIf two parallel lines are cut by a third line, the third line is called the transversal.
\nSo,
\nThe relationships among angles that are created when a line intersects two parallel lines are :
\n(1) Corresponding angles
\n(2) Vertically Opposite angles
\n(3) Alternate interior angles
\n(4) Alternate exterior angles<\/p>\n

Try It!<\/strong><\/p>\n

Which angles are congruent to \u22208?
\n\"Envision
\n\"Envision \"Envision \"Envision
\nWhich angles are supplementary to \u22208?
\n\"Envision \"Envision \"Envision \"Envision
\nAnswer:
\nThe given transversal is:
\n\"Envision
\nWe know that,
\nThe angle relationships that have equal angle measures are:
\na. Corresponding angles b. Vertically opposite angles c. Alternate interior angles d. Alternate exterior angles
\nWe know that,
\nThe angle relationships that have supplementary angle measures are:
\na. Adjacent angles
\nHence, from the above,
\nWe can conclude that
\nThe angles that are congruent to \u22208 are: \u22204, \u22206 and \u22202
\nThe angles that are supplementary to \u22208 are: \u22207, \u22201, \u22202, and \u22206<\/p>\n

Convince Me!
\nUse what you know about other angle relationships to explain why \u22204 and \u22205 are supplementary angles.
\nAnswer:
\nThe given transversal is:
\n\"Envision
\nNow,
\nFrom the given Transversal,
\nWe can observe that
\n\u22204 and \u22205 are consecutive interior angles
\nNow,
\nWe know that,
\nThe sum of the consecutive interior angles is always equal to 180\u00b0
\nThe angles that have the sum 180\u00b0 are also called “Supplementary angles”
\nHence, from the above,
\nWe can conclude that \u22204 and \u22205 are supplementary angles<\/p>\n

Try It!<\/strong><\/p>\n

What are the measures of \u22207 and \u22202? Explain.
\n\"\"
\nAnswer:
\nThe given transversal is:
\n\"\"
\nNow,
\nFrom the given transversal,
\nWe can observe that
\n\u22204 and 99\u00b0 are the corresponding angles
\n\u22204 and \u22202 are alternate interior angles
\n\u22205 and \u22207 are vertical angles
\n\u22204 and \u22205 are supplementary angles
\nSo,
\nNow,
\n\u22204 = 99\u00b0
\nNow,
\n\u22204 + \u22205 = 180\u00b0
\n\u22205 = 180\u00b0 – 99\u00b0
\n\u22205 = 81\u00b0
\nNow,
\n\u22202 = 99\u00b0
\n\u22207 = 81\u00b0
\nHence, from the above,
\nWe can conclude that
\nThe values of \u22207 and \u22202 are: 81\u00b0 and 99\u00b0 respectively<\/p>\n

Try It!<\/strong><\/p>\n

In the figure, a || b. What is the value of x?
\n\"Envision
\nAnswer:
\nIt is given that a || b
\nNow,
\nThe given transversal is:
\n\"Envision
\nFrom the given transversal,
\nWe can observe that
\n59\u00b0 and (x + 12)\u00b0 are consecutive exterior angles
\nNow,
\nWe know that,
\nThe sum of the angles of consecutive exterior angles is: 180\u00b0
\nSo,
\n59\u00b0+ (x + 12)\u00b0 = 180\u00b0
\nx + 71\u00b0 = 180\u00b0
\nx = 180\u00b0 – 71\u00b0
\nx = 109\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 109\u00b0<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

If parallel lines are intersected by a transversal, then
\n\u2022 Corresponding angles are congruent.
\n\u2022 Alternate interior angles are congruent.
\n\u2022 Same-side interior angles are supplementary.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> What are the relationships among angles that are created when a line intersects two parallel lines?
\nAnswer:
\nIf two parallel lines are cut by a third line, the third line is called the transversal.
\nSo,
\nThe relationships among angles that are created when a line intersects two parallel lines are :
\n(1) Corresponding angles
\n(2) Vertically Opposite angles
\n(3) Alternate interior angles
\n(4) Alternate exterior angles<\/p>\n

Question 2.
\nWhen parallel lines are cut by a transversal, how can you use a translation to describe how angles are related?
\nAnswer:
\nLet a and b be the parallel lines
\nNow,
\nThe given transversal is:
\n\"\"
\nNow,
\nWhen the parallel lines are cut by a transversal,
\nThe relation between angles is:
\nCorresponding angles:
\n\u22201 = \u22205, \u22202 = \u22206, \u22203 = \u22207, and \u22204 = \u22208
\nAlternate Interior angles:
\n\u22204 = \u22206, \u22203 = \u22205
\nAlternate Exterior angles:
\n\u22201 = \u22207, \u22202 = \u22208
\nVertically Opposite Angles:
\n\u22201 = \u22207, \u22202 = \u22208
\nConsecutive Interior angles:
\n\u22204 + \u22205 = 180\u00b0, \u22203 + \u22206 = 180\u00b0
\nConsecutive Exterior angles:
\n\u22201 + \u22208 = 180\u00b0, \u22207 + \u22202 = 180\u00b0<\/p>\n

Question 3.
\nHow many angles are created when two parallel lines are cut by a transversal? How many different angle measures are there?
\nAnswer:
\nWhen two parallel lines are cut by a transversal, the number of angles created are: 8
\nNow,
\nThe different angle measures are:
\na. Corresponding angles
\nb. Alternate Interior angles
\nc. Alternate Exterior angles
\nd. Vertical angles
\ne. Consecutive interior angles
\nf. Consecutive exterior angles
\ng. Adjacent angles<\/p>\n

Question 4.
\nUse Structure How can you use angle measures to tell whether two lines are parallel?
\nAnswer:
\nThe ways to prove two lines are parallel using angle measures are:
\na. If corresponding angles are congruent.
\nb. If alternate interior angles are congruent.
\nc. If consecutive, or same side, interior angles are supplementary.
\nd. If alternate exterior angles are congruent.<\/p>\n

Do You Know How?<\/strong>
\nIn 5-7, use the figure below.
\n\"Envision<\/p>\n

Question 5.
\nWhich angles are congruent to \u22208?
\nAnswer:
\nThe given transversal is:
\n\"Envision
\nNow,
\nWe know that,
\nThe angle relationships that have equal angle measures are:
\na. Corresponding angles b. Vertically opposite angles c. Alternate interior angles d. Alternate exterior angles
\nHence, from the above,
\nWe can conclude that
\nThe angles that are congruent to \u22208 are: \u22202, \u22204, and \u22206<\/p>\n

Question 6.
\nIf m\u22204 = 70\u00b0, what is m\u22206? Explain.
\nAnswer:
\nThe given transversal is:
\n\"Envision
\nNow,
\nFrom the given transversal,
\nWe can observe that
\n\u22204 and \u22206 are alternate interior angles
\nWe know that,
\nThe alternate interior angles are congruent
\nHence, from the above,
\nWe can conclude that
\n\u22204 = \u22206 = 70\u00b0<\/p>\n

Question 7.
\nIf m\u22201 = 95\u00b0, write an equation that could be used to find the measure of \u22208. Find m\u22208.
\nAnswer:
\nThe given transversal is:
\n\"Envision
\nNow,
\nFrom the given transversal,
\nWe can observe that
\n\u22201 and \u22208 are consecutive exterior angles
\nNow,
\nWe know that,
\nThe sum of the consecutive exterior angle measures is 180\u00b0
\nSo,
\n\u22201 + \u22208 = 180\u00b0
\n\u22208 = 180\u00b0 – \u22201
\n\u22208 = 180\u00b0 – 95\u00b0
\n\u22208 = 85\u00b0
\nHence, from the above,
\nWe can conclude that the measure of \u22208 is: 85\u00b0<\/p>\n

Question 8.
\nWhat must x equal if line a is parallel to line b?
\n\"Envision
\nAnswer:
\nIt is given that line a is parallel to line b
\nNow,
\nThe given transversal is:
\n\"Envision
\nNow,
\nFrom the given transversal,
\nWe can observe that
\n(2x + 35)\u00b0 and 103\u00b0 are the corresponding angles
\nWe know that,
\nThe corresponding angles are congruent
\nSo,
\n(2x + 35)\u00b0 = 103\u00b0
\n2x\u00b0 = 103\u00b0 – 35\u00b0
\n2x\u00b0 = 68\u00b0
\nx\u00b0 = \\(\\frac{68\u00b0}{2}\\)
\nx\u00b0 = 34\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 34\u00b0<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 9.
\nIf p || q, what is the value of u?
\n\"Envision
\nAnswer:
\nThe given transversal is:
\n\"Envision
\nFrom the given transversal,
\nWe can observe that
\nu and 148\u00b0 are vertically opposite angles
\nNow,
\nWe know that.
\nThe vertically opposite angles are always congruent
\nHence, from the above,
\nWe can conclude that the value of u is: 148\u00b0<\/p>\n

Question 10.
\nAre \u2220K and \u2220B corresponding angles? Explain.
\n\"Envision
\nAnswer:
\nThe given transversal is:
\n\"Envision
\nNow,
\nWe know that,
\nCorresponding angles are\u00a0angles that are in the same position relative to lines intersected by a transversal
\nSo,
\nFrom the given transversal,
\nWe can observe that
\n\u2220B is internal and \u2220K is external. They are equals if the two intersected lines by the transversal are parallel
\nHence, from the above,
\nWe can conclude that
\n\u2220K and \u2220B are called “Corresponding angles” only when m || n<\/p>\n

Question 11.
\nStreets A and B run parallel to each other. The measure of \u22206 is 155\u00b0. What is the measure of \u22204?
\n\"Envision
\nAnswer:
\nIt is given that
\nStreets A and B run parallel to each other. The measure of \u22206 is 155\u00b0
\nNow,
\nThe given representation of streets A and B are:
\n\"Envision
\nNow,
\nFrom the given representation,
\nWe can observe that
\n\u22204 and \u22206 are consecutive interior angles
\nWe know that,
\nThe sum of consecutive interior angles is always equal to 180\u00b0
\nSo,
\n\u22204 + \u22206 = 180\u00b0
\n\u22204 = 180\u00b0 – \u22206
\n\u22204 = 180\u00b0 – 155\u00b0
\n\u22204 = 25\u00b0
\nHence, from the above,
\nWe can conclude that the measure of \u22204 is: 25\u00b0<\/p>\n

Question 12.
\nReasoning<\/strong> The figure shows the design of a rectangular windowpane. The four horizontal lines are parallel. The measure of \u22206 is 53\u00b0. What is the measure of \u222012? Write and solve an equation to find the answer.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe figure shows the design of a rectangular windowpane. The four horizontal lines are parallel. The measure of \u22206 is 53\u00b0
\nNow,
\nThe representation of the rectangular windowpane is:
\n\"Envision
\nNow,
\nFrom the given representation,
\nWe can observe that
\n\u22206 and \u222012 are consecutive interior angles
\nWe know that,
\nThe sum of consecutive interior angles is: 180\u00b0
\nso,
\n\u22206 + \u222012 = 180\u00b0
\n\u222012 = 180\u00b0 – \u22206
\n\u222012 = 180\u00b0 – 53\u00b0
\n\u222012 = 127\u00b0
\nHence, from the above,
\nWe can conclude that the angle measure of \u222012 is: 127\u00b0<\/p>\n

Question 13.
\nIn the figure, m || n. If m\u22208 is (4x + 7)\u00b0 and m\u22202 is 107\u00b0, what is the value of x? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nm || n and m\u22208 is (4x + 7)\u00b0 and m\u22202 is 107\u00b0
\nNow,
\nThe given transversal is:
\n\"Envision
\nFrom the given transversal,
\nWe can observe that
\n\u22202 and \u22208 are vertically opposite angles
\nWe know that,
\nThe vertically opposite angles are always congruent
\nSo,
\n\u22202 = \u22208
\n107\u00b0 = (4x + 7)\u00b0
\n4x\u00b0 = 107\u00b0- 7\u00b0
\n4x\u00b0 = 100\u00b0
\nx\u00b0 = \\(\\frac{100\u00b0}{4}\\)
\nx\u00b0 = 25\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 25\u00b0<\/p>\n

Question 14.
\nFor the given figure, can you conclude m || n? Explain.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\n74\u00b0 and 74\u00b0 are alternate interior angles
\nNow,
\nWe know that,
\nThe Converse of the Alternate Interior Angles Theorem states that\u00a0if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel
\nHence, from the above,
\nWe can conclude that m || n since the alternate interior angles are congruent<\/p>\n

Question 15.
\nLine m is parallel to line n. Find the value of x and each missing angle measure.
\n\"Envision
\nAnswer:
\nIt is given that line m is parallel to n
\nNow,
\nThe given transversal is:
\n\"Envision
\nFrom the given transversal,
\nWe can observe that
\n(2x + 25)\u00b0 and 86\u00b0 are vertically opposite angles
\nWe know that,
\nThe vertically opposite angles are always congruent
\nSo,
\n(2x + 25)\u00b0 = 86\u00b0
\n2x\u00b0 = 86\u00b0- 25\u00b0
\n2x\u00b0= 61\u00b0
\nx\u00b0 = \\(\\frac{61\u00b0}{2}\\)
\nx\u00b0 = 30.5\u00b0
\nSo,
\nThe value of x is: 30.5\u00b0
\nNow,
\nWe can observe that
\n\u22201 and 86\u00b0 are supplementary angles
\n\u22201 and \u22203 are vertical angles
\n\u22202 and \u22203 are supplementary angles
\n(2x + 25)\u00b0\u00a0and \u22206 are supplementary angles
\n\u22206 and \u22204 are vertical angles
\n\u22204 and \u22205 are supplementary angles
\nSo,
\n\u22201 + 86\u00b0 = 180\u00b0
\n\u22201 = 180\u00b0 – 86\u00b0
\n\u22201 = 94\u00b0
\n\u22201 = \u22203 = 94\u00b0
\n\u22202 + \u22203 = 180\u00b0
\n\u22202 = 180\u00b0 – \u22203
\n\u22202 = 180\u00b0 – 94\u00b0
\n\u22202 = 86\u00b0
\nNow,
\n(2x + 25)\u00b0 + \u22206 = 180\u00b0
\n2 (30.5)\u00b0 + \u22206 = 180\u00b0 – 25\u00b0
\n\u22206 = 155\u00b0 – 61\u00b0
\n\u22206 = 94\u00b0
\n\u2220 6 = \u22204 = 94\u00b0
\n\u22204 + \u22205 = 180\u00b0
\n\u22205 = 180\u00b0 – \u22204
\n\u22205 = 180\u00b0 – 94\u00b0
\n\u22205 = 96\u00b0
\nHence, from the above,
\nWe can conclude that
\nThe value of x is: 30.5\u00b0
\nThe missing angle measures are:
\n\u22201 = \u22203 = 94\u00b0; \u22204 = \u22206 = 94\u00b0; \u22202 = 86\u00b0; \u22205 = 96\u00b0<\/p>\n

Question 16.
\nHigher Order Thinking<\/strong>
\na. Find the value of x given that r || s.
\n\"Envision
\nm\u22201 = (63 \u2013 x)\u00b0
\nm\u22202 = (72 \u2013 2x)\u00b0
\nAnswer:
\nIt is given that r || s
\nNow,
\nThe given transversal is:
\n\"Envision
\nFrom the given transversal,
\nWe can observe that
\n\u22201 and \u22202 are corresponding angles
\nSo,
\n\u22201 = \u22202
\nSo,
\n(63 – x)\u00b0 = (72 – 2x)\u00b0
\n2x\u00b0 – x\u00b0= 72\u00b0 – 63\u00b0
\nx\u00b0 = 9\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 9\u00b0<\/p>\n

b. Find m\u22201 and m\u22202.
\nAnswer:
\nFrom part (a),
\nWe can observe that the value of x is: 9\u00b0
\nSo,
\n\u22201 = (63 – x)\u00b0
\n\u22201 = 63\u00b0 – 9\u00b0
\n\u22201 = 56\u00b0
\n\u22202 = (72 – 2x)\u00b0
\n\u22202 = 72\u00b0 – 2 (9)\u00b0
\n\u22202 = 72\u00b0 – 18\u00b0
\n\u22202 = 54\u00b0
\nHence, from the above,
\nWe can conclude that
\nThe values of \u22201 and \u22202 are: 56\u00b0 and 54\u00b0 respectively<\/p>\n

Question 17.
\nFind the measures of \u2220b and \u2220d given that m || n.
\n\"Envision
\nAnswer:
\nIt is given that m || n
\nNow,
\nThe given transversal is:
\n\"Envision
\nFrom the given transversal,
\nWe can observe that
\n\u2220b and 119.3\u00b0 are supplementary angles
\nSo,
\n\u2220b + 119.3\u00b0 = 180\u00b0
\n\u2220b = 180\u00b0 – 119.3\u00b0
\n\u2220b = 70.7\u00b0
\nNow,
\n\u2220d and 136.9\u00b0 are supplementary angles
\nSo,
\n\u2220d + 136.9\u00b0 = 180\u00b0
\n\u2220d = 180\u00b0 – 136.9\u00b0
\n\u2220d = 43.1\u00b0
\nHence, from the above,
\nWe can conclude that
\nThe values of \u2220b and \u2220d are 70.7\u00b0 and 43.1\u00b0 respectively<\/p>\n

Assessment Practice<\/strong>
\nQuestion 18.
\nIn the figure, g || p. Which angles are alternate interior angles? Select all that apply.
\n\"Envision
\n\u2610 \u2220q and \u2220r
\n\u2610 \u2220q and \u2220t
\n\u2610 \u2220q and \u2220k
\n\u2610 \u2220r and \u2220t
\n\u2610 \u2220r and \u2220k
\n\u2610 \u2220u and \u22209
\nAnswer:
\nIt is given that g || p
\nNow,
\nThe given transversal is:
\n\"Envision
\nFrom the given transversal,
\nWe can observe that
\n\u2220r and \u2220k are alternate interior angles
\n\u2220q and \u2220t are alternate interior angles
\nHence, from the above,
\nWe can conclude that
\nThe alternate interior angles are:
\n\"\"<\/p>\n

Question 19.
\nIn the figure, p || q. On a recent math test, Jacob incorrectly listed the value of w as 101.
\n\"Envision<\/p>\n

PART A
\nFind the value of w.
\nAnswer:
\nIt is given that p || q
\nNow,
\nThe given transversal is:
\n\"Envision
\nFrom the given transversal,
\nWe can observe that
\n79\u00b0 and \u2220w are the corresponding angles
\nHence, from the above,
\nWe can conclude that the value of \u2220w is: 79\u00b0<\/p>\n

PART B
\nWhat mistake did Jacob likely make?
\nAnswer:
\nWe know that,
\nThe corresponding angles are always congruent
\nHence,
\nThe mistake that Jacob likely made is:
\nJacob considers \u2220w\u00a0 and 101\u00b0 the corresponding angles<\/p>\n

Lesson 6.9 Interior and Exterior Angles of Triangles<\/h3>\n

Solve & Discuss It!<\/strong>
\nNell cuts tile to make a decorative strip for a kitchen backsplash. She must cut the tiles precisely to be congruent triangles. She plans to place the tiles between two pieces of molding, as shown. What is m\u22202? Explain.
\n\"Envision
\nI can… find the interior and exterior angle measures of a triangle.
\nAnswer:
\nIt is given that
\nNell cuts tile to make a decorative strip for a kitchen backsplash. She must cut the tiles precisely to be congruent triangles. She plans to place the tiles between two pieces of molding, as shown
\nNow,
\nThe representation of the decorative strip for a kitchen backsplash is:
\n\"\"
\nNow,
\nFrom the given representation,
\nWe can observe that
\n\u22201 of second congruent triangle = \u22201 of the first congruent triangle
\n\u22203 of second congruent triangle = \u22203 of the third congruent triangle
\nNow,
\nWe know that,
\nThe sum of all the adjacent angles in a transversal is 180\u00b0
\nSo,
\n\u22201 + \u22202 + \u22203 = 180\u00b0
\nFrom the given representation,
\n\u22201 = \u22203 = 65\u00b0
\nSo,
\n\u22202 = 180\u00b0 – (65\u00b0 + 65\u00b0)
\n\u22202 = 180\u00b0 – 130\u00b0
\n\u22202 = 50\u00b0
\nHence, from the above,
\nWe can conclude that the value of \u22202 is: 50\u00b0<\/p>\n

Model with Math
\nHow can you use your knowledge of parallel lines and transversals to solve the problem?
\nAnswer:
\nThe representation of the above problem in the form of parallel lines and transversals is:
\n\"\"
\nNow,
\nFrom the above figure,
\nWe can observe that
\nAccording to the alternate interior angles theorem,
\n\u22201 of second congruent triangle = \u22201 of the first congruent triangle
\n\u22203 of second congruent triangle = \u22203 of the third congruent triangle
\nNow,
\nWe know that,
\nThe sum of all the adjacent angles in a transversal is 180\u00b0<\/p>\n

Focus on math practices
\nReasoning What assumption(s) did you need to make to find m\u22202? Explain why your assumption(s) is reasonable.
\nAnswer:
\nThe assumptions you need to find \u22202 are:
\na. \u22201 of the second congruent triangle = \u22201 of the first congruent triangle
\nb. \u22203 of the second congruent triangle = \u22203 of the third congruent triangle
\nc. The sum of all the adjacent angles in a transversal is 180\u00b0<\/p>\n

Essential Question<\/strong>
\nHow are the interior and exterior angles of a triangle related?
\nAnswer:
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.<\/p>\n

Try It!<\/strong><\/p>\n

Find the unknown angle measure in the triangle at the right.
\n\"Envision
\nAnswer:
\nThe given triangle is:
\n\"Envision
\nNow,
\nWe know that,
\nThe sum of all the interior angles in a triangle is: 180\u00b0
\nNow,
\nLet x\u00b0 be the unknown angle measure
\nSo,
\n68\u00b0 + 40\u00b0 + x\u00b0 = 180\u00b0
\nx\u00b0 = 180\u00b0 – 108\u00b0
\nx\u00b0 = 72\u00b0
\nHence, from the above,
\nWe can conclude that the value of the unknown angle measure is: 72\u00b0<\/p>\n

Convince Me!
\nCould a triangle have interior angle measures of 23\u00b0, 71\u00b0, and 96\u00b0? Explain.
\nAnswer:
\nThe given interior angle measures of a triangle are 23\u00b0, 71\u00b0, and 96\u00b0
\nNow,
\nWe know that,
\nThe sum of all the interior angles in a triangle is: 180\u00b0
\nSo,
\n23\u00b0 + 71\u00b0 + 96\u00b0 = 180\u00b0
\n94\u00b0 + 96\u00b0 = 180\u00b0
\n190\u00b0 \u2260 180\u00b0
\nHence, from the above,
\nWe can conclude that a triangle could not have given interior angle measures<\/p>\n

Try It!<\/strong><\/p>\n

What is the measure of the exterior angle shown?
\n\"Envision
\nAnswer:
\nThe given triangle is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\n(7x – 1)\u00b0 and (8x + 8)\u00b0 are interior angle measures
\n16x\u00b0 is an exterior angle measure
\nNow,
\nWe know that,
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.
\nSo,
\n16x\u00b0 = (7x – 1)\u00b0 + (8x + 8)\u00b0
\n16x\u00b0 = 15x\u00b0 + 7\u00b0
\n16x\u00b0 – 15x\u00b0 = 7\u00b0
\nx\u00b0 = 7\u00b0
\nSo,
\n16x\u00b0 = 16 (7)\u00b0
\n= 112\u00b0
\nHence, from the above,
\nWe can conclude that the exterior angle measure of the given triangle is: 112\u00b0<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

The sum of the measures of the interior angles of a triangle is 180\u00b0.
\nm\u22201 + m\u22202 + m\u22203 = 180\u00b0
\n\"Envision
\nThe measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.
\nm\u22202 + m\u22203 = m\u22204<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How are the interior and exterior angles of a triangle related?
\nAnswer:
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.<\/p>\n

Question 2.
\nReasoning<\/strong> Maggie draws a triangle with a right angle. The other two angles have equal measures. What are the possible values of the exterior angles for Maggie’s triangle? Explain.
\nAnswer:
\nIt is given that
\nMaggie draws a triangle with a right angle. The other two angles have equal measures
\nNow,
\nWe know that,
\nThe sum of the interior angles in a triangle is: 180\u00b0
\nNow,
\nLet the unknown two angle measures be x\u00b0
\nSo,
\n90\u00b0 + x\u00b0 + x\u00b0 = 180\u00b0
\n2x\u00b0 = 180\u00b0 – 90\u00b0
\n2x\u00b0= 90\u00b0
\nx\u00b0 = \\(\\frac{90\u00b0}{2}\\)
\nx\u00b0 = 45\u00b0
\nSo,
\nThe two unknown angle measures are 45\u00b0 and 45\u00b0
\nNow,
\nWe know that,
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.
\nNow,
\nLet the exterior angle measure be y\u00b0
\nSo,
\ny\u00b0 = 45\u00b0 + 45\u00b0
\ny\u00b0 = 90\u00b0
\nHence, from the above,
\nWe can conclude that the possible value for the exterior angle measure of Maggie’s triangle is: 90\u00b0<\/p>\n

Question 3.
\nBrian draws a triangle with interior angles of 32\u00b0 and 87\u00b0, and one exterior angle of 93\u00b0. Draw the triangle. Label all of the interior angles and the exterior angle.
\nAnswer:
\nIt is given that
\nBrian draws a triangle with interior angles of 32\u00b0 and 87\u00b0, and one exterior angle of 93\u00b0
\nNow,
\nWe know that,
\nThe sum of all the interior angles in a triangle is 180\u00b0
\nNow,
\nLet the unknown interior angle measure be: x\u00b0
\nSo,
\nx\u00b0+ 32\u00b0 + 87\u00b0 = 180\u00b0
\nx\u00b0 = 180\u00b0 – 119\u00b0
\nx\u00b0 = 61\u00b0
\nNow,
\nWe know that,
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.
\nNow,
\nLet the unknown exterior angle measure be y\u00b0
\nSo,
\nFrom the given interior angle measures,
\ny\u00b0 = 61\u00b0 + 32\u00b0
\nSo,
\nIn a triangle,
\nThe interior angle measures are 61\u00b0, 32\u00b0, and 87\u00b0
\nThe exterior angle measure is: 93\u00b0
\nHence,
\nThe representation of the interior angles and the exterior angle of a triangle is:
\n\"\"<\/p>\n

Do You Know How?<\/strong>
\nUse the diagram below for 4 and 5. Assume that a || b.
\n\"Envision<\/p>\n

Question 4.
\nWhat are the measures of \u22201 and \u22202? Explain.
\nAnswer:
\nIt is given that a || b
\nNow,
\nThe given transversal is:
\n\"Envision
\nFrom the given transversal,
\nWe can observe that
\n\u22201 and 37.3\u00b0 are vertically opposite angles
\nSo,
\n\u22201 = 37.3\u00b0
\nNow,
\nWe know that,
\nThe sum of all the interior angles in a triangle is 180\u00b0
\nSo,
\n\u22201 + 79.4\u00b0 + \u22202 = 180\u00b0
\n37.3\u00b0 + 79.4\u00b0 + \u22202 = 180\u00b0
\n\u22202 = 180\u00b0 – 116.7
\n\u22202 = 63.3\u00b0
\nHence, from the above,
\nWe can conclude that the measures of \u22201 and \u22202 are 37.3\u00b0 and 63.3\u00b0 respectively<\/p>\n

Question 5.
\nWhat are the measures of \u22203 and \u22204? Explain.
\nAnswer:
\nIt is given that a || b
\nNow,
\nThe given transversal is:
\n\"Envision
\nNow,
\nFrom the given transversal,
\nWe can observe that
\n\u22203 and \u22204 are the exterior angles
\nNow,
\n\u22203 = \u22202 + 79.4\u00b0
\n\u22203 = 63.3\u00b0 + 79.4\u00b0
\n\u22203 = 132.7\u00b0
\nNow,
\n\u22204 = \u22201 + 79.4\u00b0
\n\u22204 = 37.3\u00b0 + 79.4\u00b0
\n\u22204 = 116.7\u00b0
\nHence, from the above,
\nWe can conclude that the values of \u22203 and \u22204 are: 132.7\u00b0 and 116.7\u00b0<\/p>\n

Question 6.
\nIn \u25b3ABC, M\u2220A = x\u00b0, m\u2220B = (2x)\u00b0, and m\u2220C = (6x + 18)\u00b0. What is the measure of each angle?
\nAnswer:
\nIt is given that
\nIn \u0394ABC,
\nThe angle measures of all the interior angles are:
\n\u2220A = x\u00b0, \u2220B = 2x\u00b0, and \u2220C = (6x + 18)\u00b0
\nNow,
\nWe know that,
\nThe sum of all the interior angles in a triangle is 180\u00b0
\nSo,
\nx\u00b0 + 2x\u00b0 + 6x\u00b0 + 18\u00b0 = 180\u00b0
\n9x\u00b0= 180\u00b0 – 18\u00b0
\n9x\u00b0 = 162\u00b0
\nx\u00b0 = \\(\\frac{162\u00b0}{9}\\)
\nx\u00b0 = 18\u00b0
\nSo,
\n\u2220A = 18\u00b0
\n\u2220B = 2 (18\u00b0) = 36\u00b0
\n\u2220C = (6x + 18)\u00b0 = 6 (18\u00b0) + 18\u00b0 = 126\u00b0
\nHence, from the above,
\nWe can conclude that the measure of each angle is: 18\u00b0, 36\u00b0, and 126\u00b0<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nLeveled Practice For the figure shown, find m\u22201.
\n\"Envision
\nAngle 1 is an _________ angle of the triangle.
\nm\u22201 is equal to the sum of its ___________.
\nm\u22201 = _______\u00b0 + _______\u00b0
\nm\u22201 = _______\u00b0
\nAnswer:
\nThe given triangle is:
\n\"Envision
\nNow,
\nFrom the given triangle,
\nWe can observe that
\n\u22201 is the exterior angle
\n59\u00b0 and 56\u00b0 are the interior angle measures
\nNow,
\nWe know that,
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.
\nSo,
\n\u22201 = 59\u00b0 + 56\u00b0
\n\u22201 = 115\u00b0
\nHence, from the above,
\nWe can conclude that the value of \u22201 is: 115\u00b0<\/p>\n

Question 8.
\nFind m\u22201 and m\u22202.
\n\"Envision
\nAnswer:
\nThe given triangle is:
\n\"Envision
\nFrom the given triangle,
\nWe can observe that
\nThere are 2 triangles and \u22201 is the exterior angle for the second triangle
\nNow,
\nWe know that,
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.
\nSo,
\nFor the first triangle,
\n138\u00b0 = \u22201 + 18\u00b0
\n\u22201 = 138\u00b0 – 18\u00b0
\n\u22201 = 120\u00b0
\nFor the second triangle,
\n\u22201 = \u22202 + 85\u00b0
\n120\u00b0 = \u22202 + 85\u00b0
\n\u22202 = 120\u00b0 – 85\u00b0
\n\u22202 = 35\u00b0
\nHence, from the above,
\nWe can conclude that the values of \u22201 and \u22202 are: 120\u00b0 and 35\u00b0<\/p>\n

Question 9.
\nIn \u25b3ABC, what is m\u2220c?
\n\"Envision
\nAnswer:
\nThe given triangle is:
\n\"Envision
\nNow,
\nWe know that,
\nThe sum of all the interior angles in a triangle is: 180\u00b0
\nSo,
\nx\u00b0 + (4x)\u00b0 + (5x – 13)\u00b0 = 180\u00b0
\n(10x – 13)\u00b0 = 180\u00b0
\n10x\u00b0 = 180\u00b0 + 13\u00b0
\n10x\u00b0 = 193\u00b0
\nx\u00b0 = \\(\\frac{193\u00b0}{10}\\)
\nx\u00b0 = 19.3\u00b0
\nSo,
\n\u2220C = (5x – 13)\u00b0
\n\u2220C = 5 (19.3)\u00b0 – 13\u00b0
\n\u2220C = 96.5\u00b0 – 13\u00b0
\n\u2220C = 83.5\u00b0
\nHence, from the above,
\nWe can conclude that the value of \u2220C is: 83.5\u00b0<\/p>\n

Question 10.
\nIn the figure, m\u22201 = (8x + 7)\u00b0, m\u22202 = (4x + 14)\u00b0, and m\u22204 = (13x + 12)\u00b0. Your friend incorrectly says that m\u22204 = 51\u00b0. What is m\u22204? What mistake might your friend have made?
\n\"Envision
\nAnswer:
\nIt is given that
\nIn the given figure, m\u22201 = (8x + 7)\u00b0, m\u22202 = (4x + 14)\u00b0, and m\u22204 = (13x + 12)\u00b0. Your friend incorrectly says that m\u22204 = 51\u00b0
\nNow,
\nThe given figure is:
\n\"Envision
\nFrom the given figure,
\nWe can observe that
\n\u22201, \u22202, and \u22203 are the interior angles
\n\u22204 is an exterior angle
\nNow,
\nWe know that,
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.
\nThe sum of all the interior angles in a triangle is 180\u00b0
\nSo,
\n\u22201 + \u22202 + \u22203 = 180\u00b0
\n\u22204 = \u22201 + \u22202
\nSo,
\n(13x + 12)\u00b0 = (8x + 7)\u00b0 + (4x + 14)\u00b0
\n13x\u00b0 + 12\u00b0 = 12x\u00b0 + 21\u00b0
\n13x\u00b0 – 12x\u00b0 = 21\u00b0- 12\u00b0
\nx\u00b0= 9\u00b0
\nSo,
\n\u22204 = (13x + 12)\u00b0
\n\u22204 = 13 (9)\u00b0 + 12\u00b0
\n\u22204 = 117\u00b0 + 12\u00b0
\n\u22204 = 129\u00b0
\nHence, from the above,
\nWe can conclude that
\nThe value of \u22204 is: 120\u00b0
\nThe mistake mightyour friend has made is:
\nHe considered \u22204 as an interior angle and he subtracted the value of \u22204 from 180\u00b0<\/p>\n

Question 11.
\nWhat is m\u22201?
\n\"Envision
\nAnswer:
\nThe given triangle is:
\n\"Envision
\nNow,
\nFrom the given triangle,
\nWe can observe that
\n\u22201 is an exterior angle
\n26\u00b0 and 90\u00b0 are interior angles
\nNow,
\nWe know that,
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.
\nSo,
\n\u22201 = 26\u00b0 + 90\u00b0
\n\u22201 = 116\u00b0
\nHence, from the above,
\nWe can conclude that the value of \u22201 is: 116\u00b0<\/p>\n

Question 12.
\nHigher Order Thinking<\/strong> Given that m\u22201 = (16x)\u00b0, m\u22202 = (8x + 21)\u00b0, and m\u22204 = (25x + 19)\u00b0, what is an expression for m\u22203? What is m\u22203?
\n\"Envision
\nAnswer:
\nIt is given that
\nGiven that m\u22201 = (16x)\u00b0, m\u22202 = (8x + 21)\u00b0, and m\u22204 = (25x + 19)\u00b0, what is an expression for m\u22203
\nNow,
\nThe given triangle is:
\n\"Envision
\nFrom the given figure,
\nWe can observe that
\n\u22201, \u22202, and \u22203 are the interior angles
\n\u22204 is an exterior angle
\nNow,
\nWe know that,
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.
\nThe sum of all the interior angles in a triangle is 180\u00b0
\nSo,
\n\u22201 + \u22202 + \u22203 = 180\u00b0
\n\u22204 = \u22201 + \u22202
\nSo,
\n(25x + 19)\u00b0 = (16x)\u00b0 + (8x + 21)\u00b0
\n25x\u00b0 + 19\u00b0 = 24x\u00b0 + 21\u00b0
\n25x\u00b0 – 24x\u00b0 = 21\u00b0- 19\u00b0
\nx\u00b0= 2\u00b0
\nSo,
\n\u22203 = 180\u00b0 – (16 (2)\u00b0 + 8 (2)\u00b0 + 21\u00b0)
\n\u22203 = 180\u00b0 – 69\u00b0
\n\u22203 = 111\u00b0
\nHence, from the above,
\nWe can conclude that
\nThe value of \u22203 is: 111\u00b0<\/p>\n

Question 13.
\nA ramp attached to a building is being built to help with deliveries. The angle that the bottom of the ramp makes with the ground is 37.2\u00b0. Find the measure of the other acute angle.
\n\"Envision
\nAnswer:
\nIt is given that
\nA ramp attached to a building is being built to help with deliveries. The angle that the bottom of the ramp makes with the ground is 37.2\u00b0
\nNow,
\nThe representation of the ramp is:
\n\"Envision
\nFrom the above,
\nWe can observe that the ramp is in the form of a right triangle
\nNow,
\nWe know that,
\nThe sum of all the interior angles in a triangle is: 180\u00b0
\nSo,
\n37.2\u00b0 + 90\u00b0 + x\u00b0 = 180\u00b0
\n127.2\u00b0 + x\u00b0 = 180\u00b0
\nx\u00b0= 180\u00b0 – 127.2\u00b0
\nx\u00b0= 62.8\u00b0
\nHence, from the above,
\nWe can conclude that the measure of the other acute angle is: 62.8\u00b0<\/p>\n

Assessment Practice<\/strong>
\nQuestion 14.
\nThe measure of \u2220F is 110\u00b0. The measure of \u2220E is 100\u00b0. What is the measure of \u2220D?
\n\"Envision
\nA. 150\u00b0
\nB. 80\u00b0
\nC. 70\u00b0
\nD. 30\u00b0
\nAnswer:
\nIt is given that
\nThe measure of \u2220F is 110\u00b0. The measure of \u2220E is 100\u00b0
\nNow,
\nThe given transversal is:
\n\"Envision
\nFrom the given transversal,
\nWe can observe that
\n\u2220A and \u2220D are the vertical angles
\nNow,
\nWe know that,
\nThe sum of the adjacent angles is: 180\u00b0
\nSo,
\n\u2220F + \u2220C = 180\u00b0
\n\u2220B + \u2220E = 180\u00b0
\nSo,
\n\u2220C = 180\u00b0 – \u2220F
\n\u2220C = 180\u00b0- 110\u00b0
\n\u2220C = 70\u00b0
\nNow,
\n\u2220B = 180\u00b0 – \u2220E
\n\u2220B = 180\u00b0 – 100\u00b0
\n\u2220B = 80\u00b0
\nNow,
\nWe know that,
\nThe sum of all the interior angles in a triangle is: 180\u00b0
\nSo,
\n\u2220A + \u2220B + \u2220C = 180\u00b0
\n70\u00b0 + 80\u00b0+ \u2220A = 180\u00b0
\n\u2220A = 180\u00b0 – 150\u00b0
\n\u2220A = 30\u00b0
\nSo,
\n\u2220D = 30\u00b0
\nHence, from the above,
\nWe can conclude that the measure of \u2220D is: 30\u00b0<\/p>\n

Question 15.
\nIn the figure, m\u22201 = (3x + 12)\u00b0, m\u22202 = (3x + 18)\u00b0 and m\u22203 = (7x + 10)\u00b0. What is m\u22203 in degrees?
\n\"Envision
\nAnswer:
\nIt is given that
\nm\u22201 = (3x + 12)\u00b0, m\u22202 = (3x + 18)\u00b0 and m\u22203 = (7x + 10)\u00b0
\nNow,
\nThe given triangle is:
\n\"Envision
\nFrom the given triangle,
\nWe can observe that
\n\u22201 and \u22202 are the interior angles
\n\u22203 is an exterior angle
\nNow,
\nWe know that,
\nAn exterior angle of a triangle is\u00a0equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees.
\nSo,
\n\u22203 = \u22201 + \u22202
\n(7x + 10)\u00b0 = (3x + 12)\u00b0 + (3x + 18)\u00b0
\n(7x + 10)\u00b0 = (6x + 30)\u00b0
\n7x\u00b0 – 6x\u00b0 = 30\u00b0 – 10\u00b0
\nx\u00b0 = 20\u00b0
\nSo,
\n\u22203 = 7 (20)\u00b0 + 10\u00b0
\n\u22203 = 140\u00b0 + 10\u00b0
\n\u22203 = 150\u00b0
\nHence, from the above,
\nWe can conclude that the measure of \u22203 is 150\u00b0<\/p>\n

Lesson 6.10 Angle-Angle Triangle Similarity<\/span><\/p>\n

Explore It!<\/strong>
\nJustin made two flags for his model sailboat.
\n\"Envision
\nI can… use angle measures to determine whether two triangles are similar.<\/p>\n

A. Draw and label triangles to represent each flag.
\nAnswer:
\nIt is given that
\nJustin made two flags for his model sailboat
\nHence,
\nThe representation of triangles that represent each flag is:
\n\"\"<\/p>\n

B. How are the side lengths of the triangles related?
\nAnswer:
\nThe representation of triangles that represent each flag is:
\n\"\"
\nFrom the given triangles,
\nWe can observe that,
\nAfter dilation,
\nFlag A and Flag B have different side lengths that are divided (or) multiplied by some value of scale factor and the sizes of Flag A and Flag B are different
\nHence, from the above,
\nWe can conclude that
\nAfter dilation transformation,
\nThe side lengths of Flag A is greater than the side lengths of Flag B<\/p>\n

C. How are the angle measurements of the triangles related?
\nAnswer:
\nThe representation of triangles that represent each flag is:
\n\"\"
\nFrom the given triangles,
\nWe can observe that the apex vertex of Flag A is 46\u00b0 and the base vertices of Flag B are 67\u00b0 and 67\u00b0
\nNow,
\nWe know that,
\nThe sum of all the angles in a triangle is 180\u00b0
\nSo,
\nFor Flag B,
\nLet the unknown angle measure be x\u00b0
\nSo,
\n67\u00b0 + 67\u00b0 + x\u00b0 = 180\u00b0
\nx\u00b0 = 180\u00b0 – 134\u00b0
\nx\u00b0 = 46\u00b0
\nSo,
\nFor Flag B,
\nThe apex vertex is: 46\u00b0
\nNow,
\nWhen we observe two triangles,
\nThe apex vertex of Flag A and Flag B is the same
\nHence, from the above,
\nWe can conclude that the corresponding angle measurements for Flag A and Flag B are the same<\/p>\n

Focus on math practices
\nReasoning Justin makes a third flag that has sides that are shorter than the sides of the small flag. Two of the angles for each flag measure the same. Are the third angles for each flag the same measure? Explain.
\nAnswer:
\nIt is given that
\nJustin makes a third flag that has sides that are shorter than the sides of the small flag. Two of the angles for each flag measure the same
\nSo,
\nTwo of the angles for each flag are 67\u00b0 and 67\u00b0
\nSince two angle measures are the same for all the flags, the remaining third angle measure will also be the same
\nHence, from the above,
\nWe can conclude that the third angles for each flag have the same measure<\/p>\n

Essential Question<\/strong>
\nHow can you use angle measures to determine whether two triangles are similar?
\nAnswer:
\nThe Angle-Angle (AA) Criterion states that if two angles in one triangle are congruent to two angles in another triangle, the two triangles are similar triangles.
\n\"Envision
\n\u2220A \u2245 \u2220D and B \u2245 \u2220E,
\nSo,
\n\u25b3ABC – \u25b3DEF.<\/p>\n

Try It!<\/strong><\/p>\n

Is \u25b3\u03a7\u03a5\u0396 ~ \u25b3LMN?
\n\"Envision
\nm\u2220X = _______
\nm\u2220N = ________
\nThe triangles ________ similar.
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nNow,
\nFor \u0394XYZ:
\nWe know that,
\nThe sum of all the angles in a triangle is: 180\u00b0
\nSo,
\n\u2220X + \u2220Y + \u2220Z = 180\u00b0
\n\u2220X = 180\u00b0 – 92\u00b0 – 42\u00b0
\n\u2220X = 180\u00b0 – 134\u00b0
\n\u2220X = 46\u00b0
\nFor \u0394LMN:
\nWe know that,
\nThe sum of all the angles in a triangle is: 180\u00b0
\nSo,
\n\u2220L + \u2220M + \u2220N = 180\u00b0
\n\u2220N = 180\u00b0 – 92\u00b0 – 53\u00b0
\n\u2220N = 180\u00b0 – 148\u00b0
\n\u2220N = 32\u00b0
\nNow,
\nTo find whether the given triangles are similar or not,
\nFind out the angle measures of the corresponding sides
\nSo,
\nFor \u0394XYZ and \u0394LMN,
\n\u2220X = \u2220L, \u2220Y = \u2220M,\u00a0 \u2220Z = \u2220N
\nBut,
\nWe can observe that
\n\u2220X \u2260 \u2220L
\nHence, from the above,
\nWe can conclude that
\nSince the angle measure of the corresponding sides are not the same,
\n\u0394XYZ is not similar to \u0394LMN<\/p>\n

Convince Me!
\nUse what you know about transformations and parallel lines to explain why the Angle-Angle Criterion is true for all triangles.<\/p>\n

Try It!<\/strong><\/p>\n

If QR || YZ, is \u25b3XYZ ~ \u25b3XRQ? Explain.
\n\"Envision
\nAnswer:
\nIt is given that QR || YZ
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nSince the lines are parallel, the angles that corresponding to the sides will also be the same
\nHence,
\nAccording to the Side – Side criterion,
\n\u0394XYZ is similar to \u0394XRQ<\/p>\n

Try It!<\/strong><\/p>\n

Find the value of x if the two triangles are similar. Explain.
\n\"Envision
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nIt is given that two triangles are similar
\nSo,
\nAccording to Angle – Angle criterion,
\n(15x)\u00b0 = 90\u00b0
\nSo,
\nx\u00b0 = \\(\\frac{90\u00b0}{15}\\)
\nx\u00b0 = 6\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 6\u00b0<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

The Angle-Angle (AA) Criterion states that if two angles in one triangle are congruent to two angles in another triangle, the two triangles are similar triangles.
\n\"Envision
\n\u2220A \u2245 \u2220D and B \u2245 \u2220E, so \u25b3ABC – \u25b3DEF.<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How can you use angle measures to determine whether two triangles are similar?
\nAnswer:
\nThe Angle-Angle (AA) Criterion states that if two angles in one triangle are congruent to two angles in another triangle, the two triangles are similar triangles.
\n\"Envision
\n\u2220A \u2245 \u2220D and B \u2245 \u2220E,
\nSo,
\n\u25b3ABC – \u25b3DEF.<\/p>\n

Question 2.
\nConstruct Arguments<\/strong> Claire says that the AA Criterion should be called the AAA Criterion. Explain why Claire might say this. Do you agree? Explain.
\nAnswer:
\nIt is given that
\nClaire says that the AA Criterion should be called the AAA Criterion.
\nNow,
\nWe know that
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nNow,
\nAAA criterion may be reformulated as the AAA (angle-angle-angle) similarity theorem
\nAccording to the AAA criterion,
\nTwo triangles have their corresponding angles equal if and only if their corresponding sides are proportional.
\nSo,
\nFrom the above two criterions’ statements,
\nWe can observe that the definition of both the criteria are the same
\nHence, from the above,
\nWe can conclude that you can agree with Clara<\/p>\n

Question 3.
\nReasoning<\/strong> Which triangle pairs below are always similar? Explain.
\nTwo right triangles
\nTwo isosceles right triangles
\nTwo equilateral triangles
\nAnswer:
\nThe given pairs of triangles are:
\na. Two right triangles
\nb. Two isosceles right triangles
\nc. Two equilateral triangles
\nNow,
\nWe know that,
\nTwo triangles\u00a0are said to be\u00a0similar if\u00a0their corresponding angles are congruent and the corresponding sides are in proportion
\nHence, from the above,
\nWe can conclude that
\nThe triangle pairs that are similar are:
\na. Two isosceles right triangles
\nb. Two equilateral triangles<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nAre the two triangles similar? Explain.
\n\"Envision
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nFrom the given triangles,
\nWe can observe that the apex angle measure in the first triangle is 44\u00b0 and the apex angle measure in the second triangle is 46\u00b0
\nNow,
\nWe know that
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nSo,
\n44\u00b0 \u2260 46\u00b0
\nHence, from the above,
\nWe can conclude that the two triangles are not similar<\/p>\n

Question 5.
\nIs \u25b3QRS ~ \u25b3QLM? Explain.
\n\"Envision
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nNow,
\nFrom the given triangles,
\nWe can observe that
\n\u2220R from \u0394QRS = \u2220L from \u0394QLM
\nNow,
\nWe know that
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nSo,
\n90\u00b0 = 90\u00b0
\nHence, from the above,
\nWe can conclude that \u0394QRS is similar to \u0394QLM<\/p>\n

Question 6.
\nAre the triangles similar? What is the value of x?
\n\"Envision
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nNow,
\nFrom the given triangles,
\nWe can observe that
\nThe apex angle of the smaller triangle and the apex angle of the larger triangle is the same
\nNow,
\nWe know that,
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nSo,
\nThe given two triangles are similar
\nNow,
\nAccording to AAA criterion,
\n(4x)\u00b0 and 76\u00b0 are alternate angles
\nNow,
\nWe know that,
\nThe alternate angles are always congruent
\nSo,
\n(4x)\u00b0 = 76\u00b0
\nx\u00b0 = \\(\\frac{76\u00b0}{4}\\)
\nx\u00b0 = 19\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 19\u00b0<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nIs \u25b3XYZ ~ \u25b3XTU?
\n\"Envision
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nNow,
\nWe know that,
\nThe sum of all the angles in a triangle is: 180\u00b0
\nSo,
\nIn \u0394XTU,
\n103\u00b0 + \u2220T + 48\u00b0 = 180\u00b0
\n\u2220T = 180\u00b0 – 151\u00b0
\n\u2220T = 29\u00b0
\nNow,
\n\u2220T in \u0394XTU and \u2220Y in \u0394XYZ are not the same
\nNow,
\nWe know that,
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nHence, from the above,
\nWe can conclude that
\n\u0394XTU and \u0394XYZ are not similar<\/p>\n

Question 8.
\nFor what value of x are \u25b3RST and \u25b3NSP similar? Explain.
\n\"Envision
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nNow,
\nFor the given triangles to be equal,
\nWe know that,
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nSo,
\n\u2220T in \u0394RST = \u2220P in \u0394NSP
\nSo,
\n(x + 19)\u00b0 = (2x)\u00b0
\n2x\u00b0 – x\u00b0= 19\u00b0
\nx\u00b0 = 19\u00b0
\nHence, from the above,
\nWe can conclude that the value of x so that \u0394RSt and \u0394NSP are similar is: 19\u00b0<\/p>\n

Question 9.
\nIs \u25b3FGH ~ \u25b3JIH? Explain.
\n\"Envision
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nNow,
\nWe know that,
\nThe sum of all the angles in a triangle is equal to 180\u00b0
\nSo,
\nIn \u0394JIH,
\n\u2220J + \u2220I + \u2220H = 180\u00b0
\n\u2220J = 180\u00b0 – (43\u00b0 + 35\u00b0)
\n\u2220J = 180\u00b0 – 78\u00b0
\n\u2220J = 102\u00b0
\nSo,
\nFrom the given triangles,
\nWe can observe that
\n\u2220J in \u0394JIH and \u2220F in \u0394FGH are not the same
\nNow,
\nWe know that,
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nHence, from the above,
\nWe can conclude that \u0394RST and \u0394JIH are not similar<\/p>\n

Question 10.
\nAre \u25b3RST and \u25b3NSP similar? Explain.
\n\"Envision
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nNow,
\nFor the given triangles to be equal,
\nWe know that,
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nSo,
\n\u2220T in \u0394RST = \u2220P in \u0394NSP
\nSo,
\n(x + 15)\u00b0 = (2x)\u00b0
\n2x\u00b0 – x\u00b0= 15\u00b0
\nx\u00b0 = 15\u00b0
\nHence, from the above,
\nWe can conclude that the value of x so that \u0394RSt and \u0394NSP are similar is: 15\u00b0<\/p>\n

Question 11.
\nContruct Arguments Describe how to use angle relationships to decide whether any two triangles are similar.
\nAnswer:
\nAccording to AA criterion,
\nIf two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion.<\/p>\n

Question 12.
\nHigher Order Thinking<\/strong> Are the triangles shown below similar? Explain.
\n\"Envision
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nNow,
\nFrom the given triangles,
\nWe can observe that
\nThe angle measures for the smaller triangle can be obtained by dividing the angle measures of the larger triangle by 3
\nBut,
\nWe know that,
\nThe angle measures must be equal for both the triangles with the proportionate side lengths
\nNow,
\nWe know that,
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nHence, from the above,
\nWe can conclude that the given triangles are not similar<\/p>\n

Assessment Practice<\/strong>
\nQuestion 13.
\nWhich of the following statements are true? Select all that apply.
\n\"Envision
\n\u2610 \u25b3XYZ ~ \u25b3SQR
\n\u2610 \u25b3XYZ ~ \u25b3QSR
\n\u2610 \u25b3XYZ ~ \u25b3GHI
\n\u2610 \u25b3GIH ~ \u25b3SRQ
\n\u2610 \u25b3ZXY ~ \u25b3GIH
\n\u2610 \u25b3GHI ~ \u25b3SRQ
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nNow,
\nFrom the given triangles,
\nWe can observe that
\n\u2220I in \u0394GIH or \u0394HIG is equal to \u2220R in \u0394QRS or \u0394SRQ
\n\u2220G in \u0394GIH or \u0394GHI is equal to \u2220X in \u0394XYZ or \u0394XZY
\nNow,
\nWe know that,
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nHence, from the above,
\nWe can conclude that
\na. \u25b3GIH ~ \u25b3SRQ
\nb. \u25b3XYZ ~ \u25b3GHI<\/p>\n

Question 14.
\nIs \u25b3GHI ~ \u25b3QRS? Explain your reasoning.
\n\"Envision
\nAnswer:
\nThe given triangles are:
\n\"Envision
\nNow,
\nFrom the given triangles,
\nWe can observe that
\n\u2220G in \u0394GHi and \u2220Q in \u0394QRS are the same
\nNow,
\nWe know that,
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nHence, from the above,
\nWe can conclude that \u0394GHI and \u0394QRS are similar<\/p>\n

Topic 6 REVIEW<\/h3>\n

Topic Essential Question<\/strong>
\nHow can you show that two figures are either congruent or similar to one another?
\nAnswer:
\nWhen the two figures are congruent,
\na. The shapes and sizes of the two figures are the same
\nb. The side lengths and the angle measures are the same
\nc. The orientation is not the same
\nWhen the two figures are similar,
\na. The shapes and orientations are the same
\nb. The sizes may are not the same
\nc. The angle measures are the same but the side lengths are different<\/p>\n

Vocabulary Review<\/strong>
\nComplete each sentence by matching each vocabulary word to its definition. Assume pairs of lines are parallel.
\n\"Envision
\nAnswer:
\n\"\"<\/p>\n

Use Vocabulary in Writing
\nDescribe a way to show that \u25b3ABC is congruent to \u25b3DEF Use vocabulary terms from this Topic in your description.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of \u0394ABC are:
\nA (3, 5), B (6, 2), and C (2, 2)
\nThe vertices of \u0394DEF are:
\nD (-2, -1), E (-5, -4), and F (-1, -4)
\nNow,
\nThe sequence of transformations to show \u0394ABC is congruent to \u0394DEF is:
\nStep 1:
\nReflect the vertices of \u0394ABC across the y-axis
\nStep 2:
\nTranslate the image we obtained in step 1 by 1 unit right and 6 units down so that we can obtain the vertices of \u0394DEF
\nHence,
\nThe representation of the sequence of transformations to show \u0394ABC is congruent to \u0394DEF is:
\n\"\"<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

Lesson 6.1 Analyze Translations<\/strong><\/p>\n

Quick Review<\/strong>
\nA translation is a transformation that maps each point of the preimage the same distance and in the same direction.<\/p>\n

Example<\/strong>
\nTranslate \u25b3XYZ 5 units right and 3 units up.
\nAnswer:
\n\"Envision<\/p>\n

Practice<\/strong>
\nQuestion 1.
\nDraw the image after a translation 3 units left and 2 units up.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of the preimage are:
\n(-1, -3), (0.5, 2), (3.5, 2), and (5, -3)
\nNow,
\nAfter the translation of 3 units left and 2 units up,
\nThe vertices of the image will become:
\n(-1 – 3, -3 + 2), (0.5 – 3, 2 + 2), (3.5 – 3, 2 + 2), and (5 – 3, -3 + 2)
\n(-4, -1), (-2.5, 4), (0.5, 4), and (2, -1)
\nHence,
\nThe representation of the preimage and the image is:
\n\"\"<\/p>\n

Lesson 6.2 Analyze Reflections<\/strong><\/p>\n

Quick Review<\/strong>
\nReflected figures are the same distance from the line of reflection but on opposite sides.<\/p>\n

Example<\/strong>
\nWhat are the coordinates of the image of \u25b3ABC after a reflection across the y-axis?
\n\"Envision
\nAnswer:
\nUse the rule (x, y) \u2192 (-x, y).
\nA (-4, 1) \u2192 A'(4, 1)
\nB (-1, 1) \u2192 B'(1, 1)
\nC (-1, 5) \u2192 C'(1, 5)<\/p>\n

Practice<\/strong>
\nUse the figure.
\n\"Envision
\nQuestion 1.
\nWhat are the coordinates of the image of rectangle WXYZ after a reflection across the X-axis?
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nFrom the given coordinate plane,
\nThe vertices of the rectangle WXYZ are:
\nW (-4, -2), X (-1, -2), Y (-1, -4), and Z (-4, -4)
\nNow,
\nWe know that,
\nIf a point reflects across the x-axis,
\nThe x-coordinate will be the same but the y-coordinate will change the sign
\nSo,
\n(x, y) before reflection —– > (x, -y) after reflection
\nHence,
\nThe vertices of the rectangle WXYZ after reflection across the x-axis will become:
\nW’ (-4, 2), X’ (-1, 2), Y’ (-1, 4), and Z’ (-4, 4)
\nHence,
\nThe representation of the rectangle WXYZ (Preimage) and the rectangle W’X’Y’Z’ (Image) is:
\n\"\"<\/p>\n

Question 2.
\nWhat are the coordinates of the image of WXYZ after a reflection across the y-axis?
\nAnswer:
\nWe know that,
\nThe vertices of the rectangle WXYZ are:
\nW (-4, -2), X (-1, -2), Y (-1, -4), and Z (-4, -4)
\nNow,
\nWe know that,
\nIf a point reflects across the y-axis,
\nThe y-coordinate will be the same but the x-coordinate will change the sign
\nSo,
\n(x, y) before reflection —– > (-x, y) after reflection
\nHence,
\nThe vertices of the rectangle WXYZ after reflection across the y-axis will become:
\nW’ (4, -2), X’ (1, -2), Y’ (1, -4), and Z’ (4, -4)
\nHence,
\nThe representation of the rectangle WXYZ (Preimage) and the rectangle W’X’Y’Z’ (Image) is:
\n\"\"<\/p>\n

Lesson 6.3 Analyze Rotations<\/strong><\/p>\n

Quick Review<\/strong>
\nA rotation turns a figure about a fixed point, called the center of rotation. The angle of rotation is the number of degrees the figure is rotated.<\/p>\n

Example<\/strong>
\nWhat are the coordinates of the image of \u25b3ABC after a 90\u00b0 rotation about the origin?
\n\"Envision
\nAnswer:
\nUse the rule (x, y) \u2192 (-y, x).
\nA(1, 4) \u2192 A'(-4, 1)
\nB(4, 4) \u2192 B'(-4, 4)
\nC(4, 1) \u2192 C'(-1, 4)<\/p>\n

Practice<\/strong>
\nUse the figure.
\n\"Envision
\nQuestion 1.
\nWhat are the coordinates of the image of quadrilateral STUV after a 180\u00b0 rotation about the origin?
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of the quadrilateral STUV are:
\nS (-4, -2), T (-2, -2), U (-2, -4), and V (-4, -4)
\nNow,
\nWe know that,
\nWhen any point rotates 180\u00b0 counterclockwise about the origin,
\n(x, y) before rotating 180\u00b0 —– > (-x, -y) after rotating 180\u00b0
\nSo,
\nThe vertices of the quadrilateral STUV after the rotation of 180\u00b0 are:
\nS’ (4, 2), T (2, 2), U (2, 4), and V (4, 4)
\nHence,
\nThe representation of the quadrilateral STUV after the rotation of 180\u00b0 is:
\n\"\"<\/p>\n

Question 2.
\nWhat are the coordinates of the image of quadrilateral STUV after a 270\u00b0 rotation about the origin?
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of the quadrilateral STUV are:
\nS (-4, -2), T (-2, -2), U (-2, -4), and V (-4, -4)
\nNow,
\nWe know that,
\nWhen any point rotates 270\u00b0 counterclockwise about the origin,
\n(x, y) before rotating 270\u00b0 —– > (y, -x) after rotating 270\u00b0
\nSo,
\nThe vertices of the quadrilateral STUV after the rotation of 270\u00b0 are:
\nS’ (-2, 4), T (-2, 2), U (-4, 2), and V (-4, 4)
\nHence,
\nThe representation of the quadrilateral STUV after the rotation of 180\u00b0 is:
\n\"\"<\/p>\n

Lesson 6.4 Compose Transformations<\/strong><\/p>\n

Quick Review<\/strong>
\nTo compose a sequence of transformations, perform one transformation, and then use the resulting image to perform the next transformation.<\/p>\n

Example<\/strong>
\nHow can you use a sequence of transformations to map Figure A onto Figure B?
\n\"Envision
\nAnswer:
\nTranslate Figure A 3 units up, and then reflect Figure A across the y-axis.<\/p>\n

Practice<\/strong>
\nQuestion 1.
\nTranslate rectangle ABCD 5 units down, and then reflect it across the y-axis.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of the rectangle ABCD are:
\nA (-5, 4), B (-1, 4), C (-1, 2), and D (-5, 2)
\nNow,
\nThe sequence of transformations to draw the image of the rectangle ABCD is:
\nStep 1:
\nTranslate the rectangle ABCD 5 units down
\nSo,
\nThe vertices of the rectangle ABCD are:
\nA (-5, 4 – 5), B (-1, 4 – 5), C (-1, 2 – 5), and D (-5, 2 -5)
\nA (-5, -1), B (-1, -1), C (-1, -3), and D (-5, -3)
\nStep 2:
\nReflect the vertices that we obtained in step 1 across the y-axis
\nNow,
\nWe know that,
\nIf a point reflects across the y-axis,
\nThe y-coordinate will be the same but the x-coordinate will change the sign
\nSo,
\n(x, y) before reflection —– > (-x, y) after reflection
\nSo,
\nThe vertices of the rectangle ABCD are:
\nA (5, -1), B (1, -1), C (1, -3), and D (5, -3)
\nHence,
\nThe representation of the sequence of transformations to represent the rectangle ABCD and its image is:
\n\"\"<\/p>\n

Lesson 6.5 Understand Congruent Figures<\/strong><\/p>\n

Quick Review<\/strong>
\nTwo figures are congruent if a sequence of transformations maps one figure onto the other.<\/p>\n

Example<\/strong>
\nHow can you determine if Figure A is congruent to Figure B?
\n\"Envision
\nAnswer:
\nReflect Figure A across the y-axis, and then translate Figure A 6 units up and 1 unit left.<\/p>\n

Practice<\/strong>
\nQuestion 1.
\nIs quadrilateral A congruent to quadrilateral B? Explain.
\n\"Envision
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of quadrilateral A are:
\n(1, 1), (2, 4), (4, 4), and (5, 1)
\nThe vertices of quadrilateral B are:
\n(-5, -4), (-4, -1), (-3, -1), and (-2, -4)
\nNow,
\nWe know that,
\nThe two figures are said to be congruent when
\na. The shapes and sizes of the figures must be the same
\nb. The side lengths of the 2 figures must be the same
\nc. The angle measures must be the same
\nSo,
\nThe representation of the quadrilateral A and the quadrilateral B is:
\n\"\"
\nHence, from the above,
\nWe can conclude that quadrilateral A and quadrilateral B are not congruent side their side lengths are not the same<\/p>\n

Lesson 6.6 Describe Dilations<\/strong><\/p>\n

Quick Review<\/strong>
\nA dilation results in an image that is the same shape but not the same size as the preimage.<\/p>\n

Example<\/strong>
\nWhat dilation maps WXYZ to W’X’Y’Z?
\n\"Envision
\nAnswer:
\nA dilation with center at the origin and a scale factor of 2 maps WXYZ to W’X’Y’Z’.<\/p>\n

Practice<\/strong>
\nUse the figure.
\n\"Envision
\nQuestion 1.
\nWhat are the coordinates of the image of parallelogram ABCD after a dilation with center (0, 0) and a scale factor of 3?
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of the parallelogram ABCD are:
\nA (-4, 1), B (0, 1), C (1, -1), and D (-3, -1)
\nNow,
\nAfter a dilation with center (0, 0) and a scale factor of 3,
\nThe vertices of the parallelogram must be multiplied with 3
\nSo,
\nA (-4, 1) \u00d7 3, B (0, 1) \u00d7 3, C (1, -1) \u00d7 3, and D (-3, -1) \u00d7 3
\nA’ (-12, 3), B’ (0, 3), C’ (3, -3), and D’ (-9, -3)
\nHence, from the above,
\nWe can conclude that the coordinates for the image of the parallelogram ABCD after a dilation with center (0, 0) and a scale factor of 3 is:
\nA’ (-12, 3), B’ (0, 3), C’ (3, -3), and D’ (-9, -3)<\/p>\n

Question 2.
\nWhat are the coordinates of the image of parallelogram ABCD after a dilation with center (0, 0) and a scale factor of \\(\\frac{1}{2}\\)?
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of the parallelogram ABCD are:
\nA (-4, 1), B (0, 1), C (1, -1), and D (-3, -1)
\nNow,
\nAfter a dilation with center (0, 0) and a scale factor of \\(\\frac{1}{2}\\),
\nThe vertices of the parallelogram must be multiplied with \\(\\frac{1}{2}\\)
\nSo,
\nA (-4, 1) \u00d7 \\(\\frac{1}{2}\\), B (0, 1) \u00d7 \\(\\frac{1}{2}\\), C (1, -1) \u00d7 \\(\\frac{1}{2}\\), and D (-3, -1) \u00d7 \\(\\frac{1}{2}\\)
\nA’ (-2, 0.5), B’ (0, 0.5), C’ (0.5, -0.5), and D’ (-1.5, -0.5)
\nHence, from the above,
\nWe can conclude that the coordinates for the image of the parallelogram ABCD after a dilation with center (0, 0) and a scale factor of \\(\\frac{1}{2}\\) is:
\nA’ (-2, 0.5), B’ (0, 0.5), C’ (0.5, -0.5), and D’ (-1.5, -0.5)<\/p>\n

Lesson 6.7 Understand Similar Figures<\/strong><\/p>\n

Quick Review<\/strong>
\nTwo-dimensional figures are similar if there is a sequence of translations, reflections, rotations, and dilations that maps one figure onto the other figure. Similar figures have the same shape, congruent angles, and proportional side lengths.<\/p>\n

Example<\/strong>
\nIs rectangle ABCD ~ rectangle A’B’C’D?
\n\"Envision
\nAnswer:
\nAll the angles are right angles.
\n\\(\\frac{A B}{A^{\\prime} B^{\\prime}}=\\frac{B C}{B^{\\prime} C^{\\prime}}=\\frac{C D}{C^{\\prime} D^{\\prime}}=\\frac{A D}{A^{\\prime} D^{\\prime}}=\\frac{2}{1}\\) = 2
\nThe figures have congruent angle measures and proportional side lengths, so they are similar.<\/p>\n

Practice<\/strong>
\nUse the figure.
\n\"Envision
\nQuestion 1.
\nIs \u25b3ABC similar to \u25b3A’B’C’ ? Explain.
\nAnswer:
\nThe given coordinate plane is:
\n\"Envision
\nNow,
\nFrom the given coordinate plane,
\nThe vertices of \u0394ABC are:
\nA (-4, 1) B (-4, 5), and C (-1, 1)
\nThe vertices of \u0394A’B’C’ are:
\nA’ (8, 2), B’ (8, 10), and C’ (2, 2)
\nNow,
\nThe sequence of transformations to find whether \u25b3ABC is similar to \u25b3A’B’C’ or not is:
\nStep 1:
\nReflect the vertices of \u25b3ABC across the y-axis
\nStep 2:
\nDilate the vertices of \u25b3ABC we obtained in step 1 with center (0, 0) and a scale factor of 2
\nHence,
\nThe representation of the sequence of transformations to represent that \u25b3ABC is similar to \u25b3A’B’C’ is:
\n\"\"<\/p>\n

Question 2.
\nWhat sequence of transformations shows that \u25b3ABC is similar to \u25b3A’B’C’?
\nAnswer:
\nThe sequence of transformations that shows \u25b3ABC is similar to \u25b3A’B’C’ is:
\nStep 1:
\nReflect the vertices of \u25b3ABC across the y-axis
\nStep 2:
\nDilate the vertices of \u25b3ABC we obtained in step 1 with center (0, 0) and a scale factor of 2<\/p>\n

Lesson 6.8 Angles, Lines, and Transversals<\/strong><\/p>\n

Quick Review<\/strong>
\nWhen parallel lines are intersected by a transversal, corresponding angles are congruent, alternate interior angles are congruent, and same-side interior angles are supplementary.<\/p>\n

Example<\/strong>
\nIf m || n, what is the value of x?
\n\"Envision
\nAnswer:
\nm\u22203 = 45\u00b0 (5x + 25)\u00b0
\n45+ (5x + 25) = 180 – n
\nx = 22<\/p>\n

Practice<\/strong>
\nIn the figure, a || b. What is the value of x?
\n\"Envision
\nAnswer:
\nIt is given that a || b
\nNow,
\nThe given transversal is:
\n\"Envision
\nNow,
\nFrom the given transversal,
\nWe can observe that
\n(3x + 9)\u00b0 and 129\u00b0 are alternate angles
\nNow,
\nWe know that,
\nThe alternate angles are always congruent
\nSo,
\n(3x + 9)\u00b0 = 129\u00b0
\n3x\u00b0 = 129\u00b0 – 9\u00b0
\n3x\u00b0 = 120\u00b0
\nx\u00b0 = \\(\\frac{120\u00b0}{3}\\)
\nx\u00b0= 40\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 40\u00b0<\/p>\n

Lesson 6.9 Interior and Exterior Angles of Triangles<\/strong><\/p>\n

Quick Review<\/strong>
\nThe sum of the measures of the interior angles of a triangle is 180\u00b0. The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.<\/p>\n

Example<\/strong>
\nFind the missing angle measure.
\n\"Envision
\nAnswer:
\nx + 40 = 100, so x = 60<\/p>\n

Practice<\/strong>
\nQuestion 1.
\nFind the missing angle measure.
\n\"Envision
\nAnswer:
\nThe given triangle is:
\n\"Envision
\nNow,
\nFrom the given triangle,
\nThe angle measures are 48\u00b0, 102\u00b0
\nLet the unknown angle measure be x\u00b0
\nNow,
\nWe know that,
\nThe sum of all the angles in a given triangle is equal to 180\u00b0
\nSo,
\n48\u00b0+ 102\u00b0 + x\u00b0 = 180\u00b0
\nx\u00b0 = 180\u00b0 – 150\u00b0
\nx\u00b0 = 30\u00b0
\nHence, from the above,
\nWe can conclude that the missing angle measure is: 30\u00b0<\/p>\n

Question 2.
\nFind the value of x.
\n\"Envision
\nAnswer:
\nThe given triangle is:
\n\"Envision
\nNow,
\nWe know that,
\nThe angle measure of an exterior angle is equal to the sum of the remote interior angles that is opposite to the given exterior angle
\nSo,
\n3x\u00b0 + 2x\u00b0 = 115\u00b0
\n5x\u00b0 = 115\u00b0
\nx\u00b0 = \\(\\frac{115\u00b0}{3}\\)
\nx\u00b0 = 23\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 23\u00b0<\/p>\n

Lesson 6-10 Angle-Angle Triangle Similarity<\/strong><\/p>\n

Quick Review<\/strong>
\nBy the AA Criterion, if two angles in one triangle are congruent to two angles in another triangle, then the triangles are similar.<\/p>\n

Example<\/strong>
\nIs \u25b3ABC ~ \u25b3DEF? Explain.
\n\"Envision
\nAnswer:
\nm\u2220B = 180\u00b0 \u2013 90\u00b0 \u2013 37\u00b0 = 53\u00b0
\nm\u2220A = m\u2220D = 90\u00b0 and m\u2220B = m\u2220E = 53\u00b0
\nBecause two angles of the triangles are congruent, the triangles are similar by the AA Criterion.<\/p>\n

Practice<\/strong>
\nQuestion 1.
\nAB || XY. Is \u25b3ABC ~ \u25b3XYC? Explain.
\n\"Envision
\nAnswer:
\nIt is given that AB || XY
\nNow,
\nThe given triangles are:
\n\"Envision
\nNow,
\nFrom the given triangles,
\nWe can observe that
\n\u2220A in \u0394ABC = \u2220X in \u0394XYC
\nNow,
\nWe know that,
\nAA criterion tells us that two triangles are similar if two corresponding angles are equal to each other
\nHence, from the above,
\nWe can conclude that \u0394ABC is similar to \u0394XYC<\/p>\n

Question 2.
\nFind the values of x and y given that \u25b3ABC is similar to \u25b3MNC.
\n\"Envision
\nAnswer:
\nIt is given that \u0394ABC is similar to \u0394MNC
\nNow,
\nThe given triangles are:
\n\"Envision
\nNow,
\nWe know that,
\nThe sum of adjacent angles is equal to 180\u00b0
\nSo,
\n4x\u00b0 + 2x\u00b0 = 180\u00b0
\n6x\u00b0 = 180\u00b0
\nx\u00b0 = \\(\\frac{180\u00b0}{6}\\)
\nx\u00b0 = 30\u00b0
\nNow,
\nSince \u2220N = 60\u00b0,
\n\u2220A = 60\u00b0 (By AA criterion)
\nNow,
\nWe know that,
\nThe sum of all the interior angles in a triangle is equal to 180\u00b0
\nSo,
\n5y\u00b0 + 60\u00b0 + 30\u00b0 = 180\u00b0
\n5y\u00b0 = 180\u00b0 – 90\u00b0
\n5y\u00b0 = 90\u00b0
\ny\u00b0 = \\(\\frac{90\u00b0}{5}\\)
\ny\u00b0 = 18\u00b0
\nHence, from the above,
\nWe can conclude that the values of x and y are: 30\u00b0 and 18\u00b0 respectively<\/p>\n

Topic 6 Fluency Practice<\/h3>\n

Crisscrossed<\/strong>
\nSolve each equation. Write your answers in the cross-number puzzle below. Each digit, negative sign, and decimal point of your answer goes in its own box.
\n\"Envision
\nI can… solve multistep equations. \u00a9 8.EE.C.7b
\n\"Envision<\/p>\n","protected":false},"excerpt":{"rendered":"

Practice with the help of enVision Math Common Core Grade 8 Answer Key Topic 6 Congruence and Similarity regularly and improve your accuracy in solving questions. enVision Math Common Core 8th Grade Answers Key Topic 6 Congruence And Similarity Topic Essential Question How can you show that two figures are either congruent or similar to … Read more<\/a><\/p>\n","protected":false},"author":13,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[10],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/90196"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/13"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=90196"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/90196\/revisions"}],"predecessor-version":[{"id":90197,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/90196\/revisions\/90197"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=90196"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=90196"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=90196"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}