Nishanth

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Big Ideas Math Book Geometry Answer Key Chapter 5 Congruent Triangles

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• Congruent Triangles Maintaining Mathematical Proficiency -Page 229
• Congruent Triangles Mathematical Practices – Page 230
• 5.1 Angles of Triangles – Page 231
• Lesson 5.1 Angles of Triangles – Page(232-238)
• Exercise 5.1 Angles of Triangles – Page(236-238)
• 5.2 Congruent Polygons – Page 239
• Lesson 5.2 Congruent Polygons – Page(240-244)
• Exercise 5.2 Congruent Polygons – Page(243-244)
• 5.3 Proving Triangle Congruence by SAS – Page 245
• Lesson 5.3 Proving Triangle Congruence by SAS – Page(246-250)
• Exercise 5.3 Proving Triangle Congruence by SAS – Page(249-250)
• 5.4 Equilateral and Isosceles Triangles – Page 251
• Lesson 5.4 Equilateral and Isosceles Triangles – Page(252-258)
• Exercise 5.4 Equilateral and Isosceles Triangles – Page(256-258)
• 5.1 to 5.4 Quiz – Page 260
• 5.5 Proving Triangle Congruence by SSS – Page 261
• Lesson 5.5 Proving Triangle Congruence by SSS – Page(262-268)
• Exercise 5.5 Proving Triangle Congruence by SSS – Page(266-268)
• 5.6 Proving Triangle Congruence by ASA and AAS – Page 269
• Lesson 5.6 Proving Triangle Congruence by ASA and AAS – Page(270-276)
• Lesson 5.6 Proving Triangle Congruence by ASA and AAS – Page(274-276)
• 5.7 Using Congruent Triangles – Page 277
• Lesson 5.7 Using Congruent Triangles – Page(278-282)
• Exercise 5.7 Using Congruent Triangles – Page(281-282)
• 5.8 Coordinate Proofs – Page 283
• Lesson 5.8 Coordinate Proofs – Page(284-288)
• Exercise 5.8 Coordinate Proofs – Page(287-288)
• Congruent Triangles Chapter Review – Page(290-294)
• Congruent Triangles Test – Page 295
• Congruent Triangles Cumulative Assessment – Page(296-297)

Congruent Triangles Maintaining Mathematical Proficiency

Find the coordinates of the midpoint M of the segment with the given endpoints. Then find the distance between the two points.

Question 1.
P(- 4, 1) and Q(0, 7)
Answer:
The given points are:
P (-4, 1), Q (0, 7)
We know that,
The midpoint M of the segment with the 2 endpoints is:
( \(\frac{x1 + x2}{2}\), \(\frac{y1 + y2}{2}\) )
Let the give points are:
(x1, y1) and (x2, y2)
So,
By comparing the given poits,
We will get
x1 = -4, x2 = 0, y1 = 1, y2 = 7
Hence,
The midpoint M = ( \(\frac{-4 + 0}{2}\), \(\frac{1 + 7}{2}\) )
= ( \(\frac{-4}{2}\), \(\frac{8}{2}\) )
= (-2, 4)
Hence, from the above,
We can conclude that the midpoint M of the segment with the given endpoints is: (-2, 4)

Question 2.
G(3, 6) and H(9, – 2)
Answer:
The given points are:
G (3, 6), H (9, -2)
We know that,
The midpoint M of the segment with the 2 endpoints is:
( \(\frac{x1 + x2}{2}\), \(\frac{y1 + y2}{2}\) )
Let the give points are:
(x1, y1) and (x2, y2)
So,
By comparing the given poits,
We will get
x1 = 3, x2 = 9, y1 = 6, y2 = -2
Hence,
The midpoint M = ( \(\frac{3 + 9}{2}\), \(\frac{6 – 2}{2}\) )
= ( \(\frac{12}{2}\), \(\frac{4}{2}\) )
= (6, 2)
Hence, from the above,
We can conclude that the midpoint M of the segment with the given endpoints is: (6, 2)

Question 3.
U(- 1, – 2) and V(8, 0)
Answer:
The given points are:
U (-1, -2), V (8, 0)
We know that,
The midpoint M of the segment with the 2 endpoints is:
( \(\frac{x1 + x2}{2}\), \(\frac{y1 + y2}{2}\) )
Let the give points are:
(x1, y1) and (x2, y2)
So,
By comparing the given poits,
We will get
x1 = -1, x2 = 8, y1 = -2, y2 = 0
Hence,
The midpoint M = ( \(\frac{-1 + 8}{2}\), \(\frac{-2 + 0}{2}\) )
= ( \(\frac{7}{2}\), \(\frac{-2}{2}\) )
= ( \(\frac{7}{2}\), -1 )
Hence, from the above,
We can conclude that the midpoint M of the segment with the given endpoints is: ( \(\frac{7}{2}\), -1 )

Solve the equation.

Question 4.
7x + 12 = 3x
Answer:
The given equation is:
7x + 12 = 3x
So,
7x – 3x = 12
4x = 12
x = \(\frac{12}{4}\)
x = 3
Hence, from the above,
We can conclude that the value of x is: 3

Question 5.
14 – 6t = t
Answer:
The given equation is:
14 – 6t = t
So,
14 = 6t + t
7t = 14
t = \(\frac{14}{7}\)
t = 2
Hence, from the above,
We can conclude that the value of t is: 2

Question 6.
5p + 10 = 8p + 1
Answer:
The given equation is:
5p + 10 = 8p + 1
So,
5p – 8p = 1 – 10
-3p = -9
3p = 9
p = \(\frac{9}{3}\)
p = 3
Hence, from the above,
We can conclude that the value of p is: 3

Question 7.
w + 13 = 11w – 7
Answer:
The given equation is:
w + 13 = 11w – 7
So,
w – 11w = -7 – 13
-10w = -20
10w = 20
w = \(\frac{20}{10}\)
w = 2
Hence, from the above,
We can conclude that the value of w is: 2

Question 8.
4x + 1 = 3 – 2x
Answer:
The given equation is:
4x + 1 = 3 – 2x
So,
4x + 2x = 3 – 1
6x = 2
x = \(\frac{2}{6}\)
x = \(\frac{1}{3}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{1}{3}\)

Question 9.
z – 2 = 4 + 9z
Answer:
The given equation is:
z – 2 = 4 + 9z
So,
z – 9z = 4 + 2
-8z = 6
z = –\(\frac{6}{8}\)
z = –\(\frac{3}{4}\)
Hence, from the above,
We can conclude that the value of z is: –\(\frac{3}{4}\)

Question 10.
ABSTRACT REASONING
Is it possible to find the length of a segment in a coordinate plane without using the Distance Formula? Explain your reasoning.
Answer:
Yes, it is possible to find the length of a segment in a coordinate plane without using the distance formula
Since the segment is a portion of a line, we can use the graph to calculate the distance of a segment even though it would not provide accurate results.
Hence,
We use the distance formula to find the length of a segment in a coordinate plane

Congruent Triangles Mathematical Practices

Monitoring Progress

Classify each statement as a definition, a postulate, or a theorem. Explain your reasoning.

Question 1.
In a coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes is – 1.
Answer:
The given statement is:
In a coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes is – 1.
We know that,
According to the “parallel and perpendicular lines theorem”, two non-vertical lines are perpendicular if and only if the product of their slopes is -1
Hence, from the above,
We can conclude that the given statement is a Theorem

Question 2.
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
Answer:
The given statement is:
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
We know that,
According to the “Linear pair perpendicular theorem”,
When two straight lines intersect at a point and form a linear pair of congruent angles, then the lines are perpendicular
Hence, from the above,
We can conclude that the given statement is a Theorem

Question 3.
If two lines intersect to form a right angle. then the lines are perpendicular.
Answer:
The given statement is:
If two lines intersect to form a right angle. then the lines are perpendicular.
We know that,
According to the “Perpendicular lines theorem”,
When two lines intersect to form a right angle, the lines are perpendicular
Hence, from the above,
We can conclude that the given statement is a Theorem

Question 4.
Through any two points, there exists exactly one line.
Answer:
The given statement is:
Through any two points, there exists exactly one line
We know that,
Between two points, only one line can be drawn and we don’t need any proof to prove the above statement
We know that,
The statement that is true without proof to prove is called “Postulate”
Hence, from the above,
We can conclude that the given statement is a Postulate

5.1 Angles of Triangles

Exploration 1

Writing a Conjecture

Work with a partner.

a. Use dynamic geometry software to draw any triangle and label it ∆ABC.
Answer:
By using the dynamic geometry software, the triangle drawn is:

b. Find the measures of the interior angles of the triangle.
Answer:
From part (a),
We can observe that the vertices of the triangle are: A, B, and C
Let the interior angles of the vertices A, B, and C be α, β, and γ respectively
Hence,
The measures of the given triangle are:

Hence, from the above,
The measures of the interior angles are:
α = 62.1°, β = 64.1°, and γ = 53.8°

c. Find the sum of the interior angle measures.
Answer:
From part (b),
The measures of the interior angles are:
α = 62.1°, β = 64.1°, and γ = 53.8°
Hence,
The sum of the interior angles = 62.1° + 64.1° + 53.1° = 180°
Hence, from the above,
We can conclude that the sum of the interior angle measures is: 180°

d. Repeat parts (a)-(c) with several other triangles. Then write a conjecture about the sum of the measures of the interior angles of a triangle.
Answer:
The representation of the 3 different triangles and their internal angle measures is:

Hence, from the above,
We can conclude that the conjecture about the sum of the measures of the interior angles of a triangle is:
The sum of the internal angle measures of a triangle is always: 180°

CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to reason inductively about data and write conjectures.
Answer:
Inductive reasoning:
Inductive reasoning is the process of arriving at a conclusion based on a set of observations.
Inductive reasoning is used in geometry in a similar way.
Conjecture:
A statement you believe to be true based on inductive reasoning.

Exploration 2

Writing a Conjecture

Work With a partner.

a. Use dynamic geometry software to draw any triangle and label it ∆ABC.
Answer:
The triangle drawn by using the dynamic geometry software is:

Hence, from the above,
We can conclude that the vertices of the triangle are: A, B, and C

b. Draw an exterior angle at any vertex and find its measure.
Answer:
From part (a),
The vertices of the triangle are: A, B, and C
Let the external angle measures of the triangle are: α, β, and γ
Hence,
The representation of the external angle measures of the triangle are:

Hence,
From the above,
We can conclude that
The external angle measures of the triangle are:
α = 310.7°, β = 299.3°, and γ = 290°

c. Find the measures of the two nonadjacent interior angles of the triangle.
Answer:
From part (b),
The external angle measures of the triangle are:
α = 310.7°, β = 299.3°, and γ = 290°
Hence,
The representation of the non-adjacent interior angles and the external angle measures of the triangle are:

Hence, from the above,
The angle measures of two non-adjacent sides are:
α = 70°, β = 60.7°, and γ = 49.3°

d. Find the sum of the measures of the two nonadjacent interior angles. Compare this sum to the measure of the exterior angle.
Answer:
From part (b),
The external angle measures of the triangle are:
α = 310.7°, β = 299.3°, and γ = 290°
From part (c),
The measures of the two non-adjacent interior angles are:
α = 70°, β = 60.7°, and γ = 49.3°
Now,
The sum of the measures of the external angles of the triangle are:
α + β + γ = 310.7° + 299.3°+ 290°
= 900.0°
The sum of the measures of the two non-adjacent interior angles is:
α + β + γ = 70° + 60.7° + 49.3°
= 180.0
Hence, from the above,
We can conclude that the sum of the measures of the external angles is 5 times the sum of the measures of the two non-adjacent interior angles

e. Repeat parts (a)-(d) with several other triangles. Then write a conjecture that compares the measure of an exterior angle with the sum of the measures of the two nonadjacent interior angles.
Answer:

Hence, from the above,
We can conclude that
The external angle measure of a vertex for a given triangle = 360° – (Internal angle measure of a vertex that we are finding the external angle measure)
The sum of the internal angle measures of the triangle is: 180°

Communicate Your Answer

Question 3.
How are the angle measures of a triangle related?
Answer:
The angle measures of a triangle are related as shown below:
The external angle measure of a vertex for a given triangle = 360° – (Internal angle measure of a vertex that we are finding the external angle measure)
The sum of the internal angle measures of the triangle is: 180°

Question 4.
An exterior angle of a triangle measures 32° What do you know about the measures of the interior angles? Explain your reasoning.
Answer:
It is given that an exterior angle of a triangle measures 32°
We know that,
The external angle measure of a vertex for a given triangle = 360° – (Internal angle measure of a vertex that we are finding the external angle measure)
So,
32° = 360° – (The internal angle measure of 32°)
The internal angle measure of 32° = 360° – 32°
The interior angle measure of 32° = 328°
Hence, from the above,
We can conclude that the interior angle measure of a triangle for an external angle measure of 32° is: 328°

Lesson 5.1 Angles of Triangles

Monitoring Progress

Question 1.
Draw an obtuse isosceles triangle and an acute scalene triangle.
Answer:
The figures of an obtuse isosceles triangle and an acute triangle are as follows:

Question 2.
∆ABC has vertices A(0, 0), B(3, 3), and C(- 3, 3), Classify the triangle by its sides. Then determine whether it is a right triangle.
Answer:
The given points are:
A (0, 0), B (3, 3), and C (-3, 3)
and the triangle is ΔABC
We know that,
To find whether the given triangle is a right-angled triangle or not,
We have to prove,
AC² = AB² + BC²
Where,
AC is the distance between A and C points
AB is the distance between A and B points
BC is the distance between B and C points
We know that,
The distance between 2 points = √(x2 – x1)² + (y2 – y1)²
Now,
Let the given points be considered as A(x1, y1), B(x2, y2), and C( x3, y3)
So,
AB = √(3 – 0)² + (3 – 0)² = √3² + 3²
= √9 + 9 = √18
BC = √(-3 – 3)² + (3 – 3)²
= √(-6)² + 0²
= √6² = 6
AC = √(-3 – 0)² + (3 – 0)²
= √(-3)² + 3²
= √9 + 9 = √18
Now,
AC² = AB² + BC²
(√18)² = (√18)² + 6²
18 = 18 + 36
18 ≠54
Hence, from the above,
We can conclude that the given triangle is not a right-angled triangle

Question 3.
Find the measure of ∠1

Answer:
The given figure is:

We know that,
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles
From the given triangle,
The exterior angle is: (5x – 10)°
The interior angles are: 40°, 3x°, ∠1
So,
(5x – 10)° = 40° + 3x°
5x° – 3x° = 40° + 10°
2x° = 50°
x = 50° ÷ 2
x = 25°
So,
The interior angles are 40°, 3 (25)°, ∠1
= 40°, 75°, ∠1
We know that,
The sum of the interior angles of a triangle is: 180°
So,
40° + 75° + ∠1 = 180°
115° + ∠1 = 180°
∠1 = 180° – 115°
∠1 = 65°
Hence, from the above,
We can conclude that the value of ∠1 is: 65°

Question 4.
Find the measure of each acute angle.

Answer:
The given figure is:

We know that,
The sum of the interior angles in a triangle is: 180°
From the given figure,
The interior angles of the right-angled triangle are: 90°, 2x°, and (x – 6)°
So,
90° + 2x° + (x – 6)° = 180°
84°+ 3x° = 180°
3x° = 180° – 84°
3x° = 96°
x = 96° ÷ 3°
x = 32°
So,
The measure of each acute angle is 90°, 2x°, (x – 6)°
= 90°, 2(32)°, (32 – 6)°
= 90°, 64°, 26°
Hence, from the above,
We can conclude that,
The measure of each acute angle is 90°, 64°, and 26°

Exercise 5.1 Angles of Triangles

Vocabulary and Core Concept Check

Question 1.
WRITING
Can a right triangle also be obtuse? Explain our reasoning.
Answer:

Question 2.
COMPLETE THE SENTENCE
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two ____________ interior angles.
Answer:
The given statement is:
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two ____________ interior angles.
Hence,
The completed form of the given statement is:
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, classify the triangle by its sides and by measuring its angles.

Question 3.

Answer:

Question 4.

Answer:
The given figure is:

We know that,
“|” represents the “Congruent” or “Equal” in geometry
So,
From the given figure,
We can observe that all three sides of the given triangle are equal
We know that,
If a triangle has all the sides equal, then the triangle is called an “Equilateral triangle”
Hence, from the above,
We can conclude that the ΔLMN is an “Equilateral triangle”

Question 5.

Answer:

Question 6.

Answer:
The given figure is:

We know that,
If any side is not equal to each other in the triangle, then the triangle is called a “Scalene triangle”
The angle greater than 90° is called as “Obtuse angle”
An angle less than 90° is called an “Acute angle”
Hence, from the above,
We can conclude that ΔABC is an “Acute scalene triangle”

In Exercises 7-10, classify ∆ABC by its sides. Then determine whether it is a right triangle.

Question 7.
A(2, 3), B(6, 3), (2, 7)
Answer:

Question 8.
A(3, 3), B(6, 9), (6, – 3)
Answer:
The given points are:
A (3, 3), B(6, 9), and C (6, -3)
We know that,
To find whether the given triangle is a right angle or not,
We have to prove,
AC² = AB² + BC²
Where,
AC is the distance between points A and C
AB is the distance between points A and B
BC is the distance between points B and C
The slope of any one side must be equal to -1
Now,
Let the given points be
A (x1, y1), B(x2, y2), and C (x3, y3)
So,
A (x1, y1)= (3, 3), B (x2, y2) = (6, 9), and C (x3, y3) = (6, -3)
We know that,
The distance between 2 points = √(x2 – x1)² + (y2 – y1)²
So,
AB = √(6 – 3)² + (9 – 3)²
= √3² + 6²
= √9 + 36 = √45
BC = √6 – 6)² + (-3 – 9)²
= √0 + 12²
= √12² = 12
AC = √(6 – 3)² + (-3 – 3)²
= √(3)² + (-6)²
= √9 + 36 = √45
So,
From the length of the sides,
We can say that the given triangle is an Isosceles triangle,
We know that,
Slope (m) = \(\frac{y2 – y1} {x2 – x1}\)
So,
Slope of AB = \(\frac{9 – 3} {6 – 3}\)
= \(\frac{6} {3}\)
= 2
Slope of BC = \(\frac{-9 – 3} {6 – 6}\)
= \(\frac{-12} {0}\)
= Undefined
Slope of AC = \(\frac{-3 – 3} {6 – 3}\)
= \(\frac{-6} {3}\)
= -2
Hence, from the above,
We can conclude that the given triangle is not a right triangle

Question 9.
A(1, 9), B(4, 8), C(2, 5)
Answer:

Question 10.
A(- 2, 3), B(0, – 3), C(3, – 2)
Answer:
The given points are:
A (-2, 3), B(0, -3), and C (3, -2)
We know that,
To find whether the given triangle is a right angle or not,
We have to prove,
AC² = AB² + BC²
Where,
AC is the distance between points A and C
AB is the distance between points A and B
BC is the distance between points B and C
The slope of any one side must be equal to -1
Now,
Let the given points be
A (x1, y1), B(x2, y2), and C (x3, y3)
So,
A (x1, y1)= (-2, 3), B (x2, y2) = (0, -3), and C (x3, y3) = (3, -2)
We know that,
The distance between 2 points = √(x2 – x1)² + (y2 – y1)²
So,
AB = √(0 – [-2])² + (3 – 3)²
= √2² + 0²
= √4 + 0 = 2
BC = √3 – 0)² + (-2 -[-3] )²
= √9 + 1²
= √10
AC = √(3 – [-2])² + (-2 – 3)²
= √(5)² + (-5)²
= √25 + 25 = √50
Now,
AC² = AB² + BC²
50 = 10 + 4
50 ≠ 14
So,
From the length of the sides,
We can say that the given triangle is a scalene triangle since all the lengths of the sides are different
We know that,
Slope (m) = \(\frac{y2 – y1} {x2 – x1}\)
So,
Slope of AB = \(\frac{9 – 3} {6 – 3}\)
= \(\frac{6} {3}\)
= 2
Slope of BC = \(\frac{-9 – 3} {6 – 6}\)
= \(\frac{-12} {0}\)
= Undefined
Slope of AC = \(\frac{-3 – 3} {6 – 3}\)
= \(\frac{-6} {3}\)
= -2
Hence, from the above,
We can conclude that the given triangle is not a right triangle

In Exercises 11 – 14. find m∠1. Then classify the triangle by its angles

Question 11.

Answer:

Question 12.

Answer:
The given figure is:

We know that,
The sum of interior angles in a triangle is: 180°
So,
From the above,
The interior angles of the given triangle are: 40°,  30°,  ∠1
Now,
40° + 30° + ∠1 = 180°
70 + ∠1 = 180°
∠1 = 180° – 70°
∠1 = 110°
We know that,
The angle greater than 90° is called an “Obtuse angle”
Hence, from the above,
We can conclude that the given triangle is an “Obtuse angled triangle”

Question 13.

Answer:

Question 14.

Answer:
The given figure is:

We know that,
The sum of interior angles in a triangle is: 180°
So,
From the above,
The interior angles of the given triangle are: 60°,  60°,  ∠1
Now,
60° + 60° + ∠1 = 180°
120 + ∠1 = 180°
∠1 = 180° – 120°
∠1 = 60°
We know that,
An angle less than 90° is called an “Acute angle”
The triangle that all the angles 60° is called an “Equilateral triangle”
Hence, from the above,
We can conclude that the given triangle is an “Equilateral triangle”

In Exercises 15-18, find the measure of the exterior angle.

Question 15.

Answer:

Question 16.

Answer:
The given figure is:

We know that,
An exterior angle is equal to the sum of the two non-adjacent interior angles in a triangle
So,
(2x – 2)° = x° + 45°
2x° – x° = 45° + 2°
x = 47°
Hence,
The measure of the exterior angle is: (2x – 2)°
= (2 (47) – 2)°
= (94 – 2)°
= 92°
Hence, from the above,
We can conclude that the measure of the exterior angle is: 92°

Question 17.

Answer:

Question 18.

Answer:
The given figure is:

We know that,
An exterior angle is equal to the sum of the two non-adjacent interior angles in a triangle
So,
(7x – 16)° = (x + 8)° + 4x°
7x° – 5x° = 16° + 8°
2x = 24°
x = 24° ÷ 2
x = 12°
Hence,
The measure of the exterior angle is: (7x – 16)°
= (7 (12) – 16)°
= (84 – 16)°
= 68°
Hence, from the above,
We can conclude that the measure of the exterior angle is: 68°

In Exercises 19-22, find the measure of each acute angle.

Question 19.

Answer:

Question 20.

Answer:
The given figure is:

From the given figure,
We can observe that one angle is 90° and the 2 sides are perpendicular
So,
We can say that the given triangle is a right-angled triangle
We know that,
The sum of interior angles of a triangle is: 180°
So,
x° + (3x + 2)° + 90° = 180°
4x° + 2° + 90° = 180°
4x° = 180° – 90° – 2°
4x° = 88°
x = 88° ÷ 4°
x = 22°
So,
The 2 acute angle measures are: x° and (3x + 2)°
= 22° and (3(22) + 2)°
= 22° and (66 + 2)°
= 22° and 68°
Hence, from the above,
We can conclude that the 2 acute angle measures are: 22° and 68°

Question 21.

Answer:

Question 22.

Answer:
The given figure is:

From the given figure,
We can observe that one angle is 90° and the 2 sides are perpendicular
So,
We can say that the given triangle is a right-angled triangle
We know that,
The sum of interior angles of a triangle is: 180°
So,
(19x – 1)° + (13x – 5)° + 90° = 180°
32x° – 6° + 90° = 180°
32x° = 180° – 90° – 6°
4x° = 84°
x = 84° ÷ 4°
x = 21°
So,
The 2 acute angle measures are: (19x – 1)° and (13x – 5)°
= (19 (21) – 1)° and (13(21) – 5)°
= 398° and (273 – 5)°
= 398° and 268°
Hence, from the above,
We can conclude that the 2 acute angle measures are: 398° and 268°

In Exercises 23-26. find the measure of each acute angle in the right triangle.

Question 23.
The measure of one acute angle is 5 times the measure of the other acute angle.
Answer:

Question 24.
The measure of one acute angle is times the measure of the other acute angle.
Answer:

Question 25.
The measure of one acute angle is 3 times the sum of the measure of the other acute angle and 8.
Answer:

Question 26.
The measure of one acute angle is twice the difference of the measure of the other acute angle and 12.
Answer:
The given statement is:
The measure of one acute angle is twice the difference of the measure of the other acute angle and 12.
So,
x° + [2 (x – 12)]° = 90°
x° + 2x° – 2(12)° = 90°
3x° – 24° = 90°
3x° = 90° + 24°
3x° = 114°
x = 114° ÷ 3
x = 38°
So,
The 2 acute angle measures are: x°, 2 (x – 12)°
= 38°, 2 (38 – 12)°
= 38°, 2(26)°
= 38° , 52°
Hence, from the above,
We can conclude that the acute angle measures are: 38°, 52°

ERROR ANALYSIS
In Exercises 27 and 28, describe and correct the error in finding m∠1.

Question 27.

Answer:

Question 28.

Answer:
We know that,
The exterior angle of a triangle is equal to the sum of the non-adjacent interior angles of a triangle
So,
From the figure,
The external angle is: ∠1
The interior angles are 80°, 50°
So,
∠1 = 80° + 50°
∠1 = 130°
Now,
The interior angle measure of ∠1= 180° – (External angle measure of 130°)
= 180° – 130°
= 50°
Hence, from the above,
The internal angle measure of ∠1 is: 50°

In Exercises 29-36, find the measure of the numbered angle.

Question 29.
∠1
Answer:

Question 30.
∠2
Answer:
We know that,
The external angle measure is equal to the sum of the non-adjacent interior angles
So,
∠2 = 90° + 40°
∠2 = 130°

Question 31.
∠3
Answer:

Question 32.
∠3
Answer:
From the above figure,
∠2 = ∠4
Hence, from the above,
We can conclude that
∠2 = ∠4 = 130°

Question 33.
∠5
Answer:

Question 34.
∠6
Answer:
The external angle measure is equal to the sum of the non-adjacent interior angles
So,
∠6 = 90° + ∠3
∠6 = 90° + 50°
∠6 = 140°

Question 35.
∠7
Answer:

Question 36.
∠8
Answer:
The external angle measure is equal to the sum of the non-adjacent interior angles
So,
∠8 = 90° + ∠1
∠6 = 90° + 50°
∠6 = 140°

Question 37.
USING TOOLS
Three people are standing on a stage. The distances between the three people are shown in the diagram. Classify the triangle by its sides and by measuring its angles.

Answer:

Question 38.
USING STRUCTURE
Which of the following sets of angle measures could form a triangle? Select all that apply.
(A) 100°, 50°, 40°
Answer:
The given angles are: 100°, 50°, 40°
We know that,
The sum of the angles of a triangle should be equal to 180°
So,
The sum of the given angles = 100° + 50° + 40°
= 100° + 90°
= 190°
Hence, from the above,
We can conclude that the given angles do not form a triangle

(B) 96°, 74°, 10°
Answer:
The given angles are: 96°, 74°, 10°
We know that,
The sum of the angles of a triangle should be equal to 180°
So,
The sum of the given angles = 96° + 74° + 10°
= 96° + 84°
= 180°
Hence, from the above,
We can conclude that the given angles forms a triangle

(C) 165°, 113°, 82°
Answer:
The given angles are: 165°, 113°, 82°
We know that,
The sum of the angles of a triangle should be equal to 180°
So,
The sum of the given angles = 165° + 113° + 82°
= 165° + 195°
= 360°
But,
We know that,
The sum of exterior angles of a triangle is: 360°
Hence, from the above,
We can conclude that the given angles forms a triangle

(D) 101°, 41°, 38°
Answer:
The given angles are: 101°, 41°, 38°
We know that,
The sum of the angles of a triangle should be equal to 180°
So,
The sum of the given angles = 101° + 38° + 41°
= 101° + 79°
= 180°
Hence, from the above,
We can conclude that the given angles forms a triangle

(E) 90°, 45°, 45°
Answer:
The given angles are: 90°, 45°, 45°
We know that,
The sum of the angles of a triangle should be equal to 180°
So,
The sum of the given angles = 90° + 45° + 45°
= 90° + 90°
= 180°
Hence, from the above,
We can conclude that the given angles forms a triangle

(F) 84°, 62°, 34°
Answer:
The given angles are: 84°, 62°, 34°
We know that,
The sum of the angles of a triangle should be equal to 180°
So,
The sum of the given angles = 84° + 62° + 34°
= 84° + 96°
= 180°
Hence, from the above,
We can conclude that the given angles forms a triangle

Question 39.
MODELING WITH MATHEMATICS
You are bending a strip of metal into an isosceles triangle for a sculpture. The strip of metal is 20 inches long. The first bend is made 6 inches from one end. Describe two ways you could complete the triangle.
Answer:

Question 40.
THOUGHT-PROVOKING
Find and draw an object (or part of an object) that can be modeled by a triangle and an exterior angle. Describe the relationship between the interior angles of the triangle and the exterior angle in terms of the object.
Answer:

From the above figure,
We can say that
The sum of the interior angles of a given triangle is: 180°
The sum of the exterior angles of a given triangle is: 360°
The relation between the interior angles and the exterior angles is:
The exterior angle measure = Sum of the two non-adjacent interior angles

Question 41.
PROVING A COROLLARY
Prove the Corollary to the Triangle Sum Theorem (Corollary 5. 1).
Given ∆ABC is a right triangle
Prove ∠A and ∠B are complementary

Answer:

Question 42.
PROVING A THEOREM
Prove the Exterior Angle Theorem (Theorem 5.2).
Given ∆ABC, exterior ∠ACD
Prove m∠A + m∠B = m∠ACD

Answer:
It is given that
In ΔABC, the exterior angle is ∠ACD
We have to prove that
m∠A + m∠B = m∠ACD
Proof:

Hence, from the above,
We can conclude that
m∠A + m∠B = m∠ACD is proven

Question 43.
CRITICAL THINKING
Is it possible to draw an obtuse isosceles triangle? obtuse equilateral triangle? If so, provide examples. If not, explain why it is not possible.
Answer:

Question 44.
CRITICAL THINKING
Is it possible to draw a right isosceles triangle? right equilateral triangle? If so, provide an example. If not, explain why it is not possible.
Answer:
It is possible to draw a right isosceles triangle but it is not possible to draw a right equilateral triangle
We know that,
In a triangle, if the length of the 2 sides are equal and one angle is a right-angle, then, it is called an “Right Isosceles triangle”
In a triangle, if the length of all the sides are equal and each angle is 60°, then it is an “Equilateral triangle”
Hence,
From the above definitions,
We can observe that it is possible to draw right isosceles triangle but it is not possible to dran a right equilateral triangle

Question 45.
MATHEMATICAL CONNECTIONS
∆ABC is isosceles.
AB = x, and BC = 2x – 4.
a. Find two possible values for x when the perimeter of ∆ABC is 32.
b. How many possible values are there for x when the perimeter of ∆ABC is 12?
Answer:

Question 46.
HOW DO YOU SEE IT?
Classify the triangles, in as many ways as possible. without finding any measurements.
a.
Answer:
The given figure is:

From the figure,
We can observe that all the length of the sides of the triangle are equal
We know that,
The triangle that has the length of all the sides equal is called an “Equilateral triangle”
Hence, from the above,
We can conclude that the given triangle is an “Equilateral triangle”

b.
Answer:
The given figure is:

From the figure,
We can observe that the lengths of all the 3 sides are different
We know that,
The triangle that has all the different side lengths is called a “Scalene triangle”
Hence, from the above,
We can conclude that the given triangle is called a “Scalene triangle”

c.
Answer:
The given figure is:

From the figure,
We can observe that the length of all the 3 sides are different and 1 angle is obtuse i.e., greater than 90°
We know that,
The triangle that has any angle obtuse is called an “Obtuse angled triangle”
Hence, from the above,
We can conclude that the given triangle is an “Obtuse angled scalene triangle”

d.
Answer:
The given figure is:

From the figure,
We can observe that 1 angle is 90° and the 2 sides are perpendicular to each other
We know that,
The triangle that has an angle of 90° and the slope -1 is called a “Right-angled triangle”
Hence, from the above,
We can conclude that the given triangle is called a “Right-angled triangle”

Question 47.
ANALYZING RELATIONSHIPS
Which of the following could represent the measures of an exterior angle and two interior angles of a triangle? Select all that apply.
A) 100°, 62°, 38°
(B) 81°, 57°, 24°
(C) 119°, 68°, 49°
(D) 95°, 85°, 28°
(E) 92°, 78°, 68°
(F) 149°, 101°, 48°
Answer:

Question 48.
MAKING AN ARGUMENT
Your friend claims the measure of an exterior angle will always be greater than the sum of the nonadjacent interior angle measures. Is your friend correct? Explain your reasoning.
Answer:
No, your friend is not correct

Explanation:
We know that,
According to the exterior angle theorem,
The external angle measure is always equal to the sum of the non-adjacent internal angle measures
But,
According to your friend,
The external angle measure will always be greater than the sum of the non-adjacent interior angle measures
Hence, from the above,
We can conclude that your friend is not correct

MATHEMATICAL CONNECTIONS
In Exercises 49-52, find the values of x and y.

Question 49.

Answer:

Question 50.

Answer:
The given figure is:

From the figure,
We have to obtain the values of x and y
Now,
By using the alternate angles theorem,
x = 118°
Now,
By using the exterior angle theorem,
x = y + 22°
y = x – 22°
y = 118° – 22°
y = 96°
Hence, from the above,
We can conclude that the values of x and y are: 118° and 96° respectively

Question 51.

Answer:

Question 52.

Answer:
The given figure is:

From the above figure,
We have to find the values of x and y
Now,
By using the sum of interior angle measures,
x° + 64° + 90° = 180°
x° + 154° = 180°
x° = 180° – 154°
x° = 26°
Now,
By using the exterior angle theorem,
y° = x° + 64°
y° = 26° + 64°
y° = 90°
Hence, from the above,
We can conclude that the values of x and y are: 26° and 90° respectively

Question 53.
PROVING A THEOREM
Use the diagram to write a proof of the Triangle Sum Theorem (Theorem 5. 1). Your proof should be different from the proof of the Triangle Sum Theorem shown in this lesson.

Answer:

Maintaining Mathematical Proficiency

Use the diagram to find the measure of the segment or angle.

Question 54.
m∠KHL
Answer:
From the given figures,
We can observe that
∠ABC = ∠GHK
∠KHL = ∠GHK / 2
So,
(6x + 2)° = (3x + 1)° + (5x – 27)°
6x – 3x – 5x = 1 – 27 – 2
6x – 8x = -27 – 1
-2x = -28
2x = 28
x = 28 ÷ 2
x = 14
So,
∠KHL = ∠GHK / 2
= [(3 (14) + 1)° + (5 (14) – 27)°] / 2
= [43° + 43°] / 2
= 86° / 2
= 43°
Hence, from the above,
We can conclude that
∠KHL = 43°

Question 55.
m∠ABC
Answer:

Question 56.
GH
Answer:
From the given figures,
We can observe that
AB = GH
So,
3y = 5y – 8
3y – 5y = -8
-2y = -8
2y = 8
y = 8 ÷ 2
y = 4
So,
The value of GH = 3y = 3 (4) = 12
Hence, from the above,
We can conclude that the value of GH is: 12

Question 57.
BC
Answer:

5.2 Congruent Polygons

Exploration 1

Describing Rigid Motions

Work with a partner: of the four transformations you studied in Chapter 4, which are rigid motions? Under a rigid motion. why is the image of a triangle always congruent to the original triangle? Explain your reasoning.

Answer:
Rigid motion occurs in geometry when an object moves but maintains its shape and size, which is unlike non-rigid motions, such as dilations, in which the object’s size changes. All rigid motion starts with the original object, called the pre-image, and results in the transformed object, called the image.
There are 4 types of rigid motion. They are:
a. Translation
b. Rotation
c. Reflection
d. Glide reflection
We know that,
Rotation only occurs in terms of 90° or 180°
Now,
The given transformations are:

So,
From the above figure,
The first figure and the second figure are different
The second figure and the third figure are the same in shape
The first figure and the fourth figure are the same in shape
So,
We can say that the first and the fourth figures are rigid motions
W can say that the second and the third figures are rigid motions
In the second and the third figures,
The “Rotation” takes place i.e., the second figure is rotated 180° keeping the original shape
In the first and the fourth figures,
The “Reflection” takes place i.e., the first figure is reflected keeping the original shape
Now,
The image of the triangle is always congruent to the original triangle because of the “Translation” i.e., the original triangle and the image of the triangle have the same sides and the same angles but not in the same position.

Exploration 2

Finding a Composition of Rigid Motions

Work with a partner. Describe a composition of rigid motions that maps ∆ABC to ∆DEF. Use dynamic geometry software to verify your answer.
LOOKING FOR STRUCTURE
To be proficient in math, you need to look closely to discern a pattern or structure.

a. ∆ABC ≅ ∆DEF

Answer:

b. ∆ABC ≅ ∆DEF

Answer:

c. ∆ABC ≅ ∆DEF

Answer:

d. ∆ABC ≅ ∆DEF

Answer:

Communicate Your Answer

Question 3.
Given two congruent triangles. how can you use rigid motions to map one triangle to the other triangle?
Answer:

Question 4.
The vertices of ∆ABC are A(1, 1), B(3, 2), and C(4, 4). The vertices of ∆DEF are D(2, – 1), E(0, 0), and F(- 1, 2). Describe a composition of rigid motions that maps ∆ABC to ∆DEF.
Answer:

Lesson 5.2 Congruent Polygons

Monitoring Progress

In the diagram, ABGH ≅ CDEF.

Question 1.
Identify all pairs of congruent corresponding parts.
Answer:

Question 2.
Find the value of x.
Answer:

Question 3.
In the diagram at the left. show that ∆PTS ≅ ∆RTQ.

Answer:

Use the diagram.

Question 4.
Find m∠DCN.
Answer:

Question 5.
What additional information is needed to conclude that ∆NDC ≅ ∆NSR?
Answer:

Exercise 5.2 Congruent Polygons

Question 1.
WRITING
Based on this lesson. what information do you need to prove that two triangles are congruent? Explain your reasoning.
Answer:

Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

Is ∆ABC ≅ ∆RST?
Answer:

Is ∆KJL ≅ ∆SRT?
Answer:

Is ∆JLK ≅ ∆STR?
Answer:

Is ∆LKJ ≅ ∆TSR?
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4. identify all pairs of congruent corresponding parts. Then write another congruence statement for the polygons.

Question 3.
∆ABC ≅ ∆DEF

Answer:

Question 4.
GHJK ≅ ∆QRST

Answer:

In Exercises 5-8, ∆XYZ ≅ ∆MNL. Copy and complete the statement.

Question 5.
m∠Y = ______
Answer:

Question 6.
m∠M = ______
Answer:

Question 7.
m∠Z = _______
Answer:

Question 8.
XY= _______
Answer:

In Exercises 9 and 10. find the values of x and y.

Question 9.
ABCD ≅ EFGH

Answer:

Question 10.
∆MNP ≅ ∆TUS

Answer:

In Exercises 11 and 12. show that the polygons are congruent. Explain your reasoning.

Question 11.

Answer:

Question 12.

Answer:

In Exercises 13 and 14, find m∠1.

Question 13.

Answer:

Question 14.

Answer:

Question 15.
PROOF
Triangular postage stamps, like the ones shown, are highly valued by stamp collectors. Prove that ∆AEB ≅ ∆CED.

Given \(\overline{A B}\) || \(\overline{D C}\), \(\overline{A B}\) ≅ \(\overline{D C}\) is the midpoint of \(\overline{A C}\) and \(\overline{B D}\)
Prove ∆AEB ≅ ∆CED
Answer:

Question 16.
PROOF
Use the information in the figure to prove that ∆ABG ≅ ∆DCF

Answer:

ERROR ANALYSIS
In Exercises 17 and 18, describe and correct the error.

Question 17.

Answer:

Question 18.

Answer:

Question 19.
PROVING A THEOREM
Prove the Third Angles Theorem (Theorem 5.4) by using the Triangle Sum Theorem (Theorem 5. 1).
Answer:

Question 20.
THOUGHT PROVOKING
Draw a triangle. Copy the triangle multiple times to create a rug design made of congruent triangles. Which property guarantees that all the triangles are congruent?
Answer:

Question 21.
REASONING
∆JKL is congruent to ∆XYZ Identify all pairs of congruent corresponding parts.
Answer:

Question 22.
HOW DO YOU SEE IT?
In the diagram, ABEF ≅ CDEF

a. Explain how you know that \(\overline{B E}\) ≅ \(\overline{D E}\) and ∠ABE ≅∠CDE.
Answer:

b. Explain how you know that ∠GBE ≅ ∠GDE.
Answer:

c. Explain how you know that ∠GEB ≅ ∠GED.
Answer:

d. Do you have enough information to prove that ∠BEG ≅ ∠DEG? Explain.
Answer:

MATHEMATICAL CONNECTIONS
In Exercises 23 and 24, use the given information to write and solve a system of linear equations to find the values of x and y.

Question 23.
∆LMN ≅ ∆PQR. m∠L = 40°, m∠M = 90° m∠P = (17x – y)°. m∠R (2x + 4y)°
Answer:

Question 24.
∆STL ≅ ∆XYZ, m∠T = 28°, m∠U = (4x + y)°, m∠X = 130°, m∠Y = (8x – 6y)°
Answer:

Question 25.
PROOF
Prove that the criteria for congruent triangles in this lesson is equivalent to the definition of congruence in terms of rigid motions.
Answer:

Maintaining Mathematical Proficiency

What can you conclude from the diagram?

Question 26.

Answer:

Question 27.

Answer:

Question 28.

Answer:

Question 29.

Answer:

5.3 Proving Triangle Congruence by SAS

Exploration 1

Drawing Triangles

Work with a partner.

Use dynamic geometry software.
a. Construct circles with radii of 2 units and 3 units centered at the origin. Construct a 40° angle with its vertex at the origin. Label the vertex A.
Answer:

b. Locate the point where one ray of the angle intersects the smaller circle and label this point B. Locate the point where the other ray of the angle intersects the larger circle and label this point C. Then draw ∆ABC.
Answer:

c. Find BC, m∠B, and m∠C.
Answer:

d. Repeat parts (a)-(c) several times. redrawing the angle indifferent positions. Keep track of your results by copying and completing the table below. What can you conclude?
USING TOOLS STRATEGICALLY
To be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data.
Answer:

Communicate Your Answer

Question 2.
What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent?
Answer:

Question 3.
How would you prove your conclusion in Exploration 1(d)?
Answer:

Lesson 5.3 Proving Triangle Congruence by SAS

Monitoring Progress

In the diagram, ABCD is a square with four congruent sides and four right
angles. R, S, T, and U are the midpoints of the sides of ABCD. Also, \(\overline{R T}\) ⊥ \(\overline{S U}\) and \(\overline{S V}\) ≅ \(\overline{V U}\).

Question 1.
Prove that ∆SVR ≅ ∆UVR.
Answer:

Question 2.
Prove that ∆BSR ≅ ∆DUT.
Answer:

Question 3.
You are designing the window shown in the photo. You want to make ∆DRA congruent to ∆DRG. You design the window so that \(\overline{D A}\) ≅ \(\overline{D G}\) and ∠ADR ≅ ∠GDR. Use the SAS Congruence Theorem to prove ∆DRA ≅ ∆DRG.

Answer:

Exercise 5.3 Proving Triangle Congruence by SAS

vocabulary and core concept check

Question 1.
WRITING
What is an included angle?
Answer:

Question 2.
COMPLETE THE SENTENCE
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then __________ .
Answer:

Monitoring progress and Modeling with Mathematics

In Exercises 3-8, name the included an1e between the pair of sides given.

Question 3.
\(\overline{J K}\) and \(\overline{K L}\)
Answer:

Question 4.
\(\overline{P K}\) and \(\overline{L K}\)
Answer:

Question 5.
\(\overline{L P}\) and \(\overline{L K}\)
Answer:

Question 6.
\(\overline{J L}\) and \(\overline{J K}\)
Answer:

Question 7.
\(\overline{K L}\) and \(\overline{J L}\)
Answer:

Question 8.
\(\overline{K P}\) and \(\overline{P L}\)
Answer:

In Exercises 9-14, decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem (Theorem 5.5). Explain.

Question 9.
∆ABD, ∆CDB

Answer:

Question 10.
∆LMN, ∆NQP

Answer:

Question 11.
∆YXZ, ∆WXZ

Answer:

Question 12.
∆QRV, ∆TSU

Answer:

Question 13.
∆EFH, ∆GHF

Answer:

Question 14.
∆KLM, ∆MNK

Answer:

In Exercises 15 – 18, write a proof.

Question 15.
Given \(\overline{P Q}\) bisects ∠SPT, \(\overline{S P}\) ≅ \(\overline{T P}\)
Prove ∆SPQ ≅ ∆TPQ

Answer:

Question 16.
Given \(\overline{A B}\) ≅ \(\overline{C D}\), \(\overline{A B}\) || \(\overline{C D}\)
Prove ∆ABC ≅ ∆CDA

Answer:

Question 17.
Given C is the midpoint of \(\overline{A E}\) and \(\overline{B D}\)
Prove ∆ABC ≅ ∆EDC

Answer:

Question 18.
Given \(\overline{P T}\) ≅ \(\overline{R T}\), \(\overline{Q T}\) ≅ \(\overline{S T}\)
Prove ∆PQT ≅ ∆RST

Answer:

In Exercises 19-22, use the given information to name two triangles that are congruent. Explain your reasoning.

Question 19.
∠SRT ≅ ∠URT, and R is the center of the circle.

Answer:

Question 20.
ABCD is a square with four congruent sides and four congruent angles.

Answer:

Question 21.
RSTUV is a regular pentagon.

Answer:

Question 22.
\(\overline{M K}\) ⊥ \(\overline{M N}\), \(\overline{K L}\) ⊥ \(\overline{N L}\), and M and L are centers of circles.

Answer:

CONSTRUCTION
In Exercises 23 and 24, construct a triangle that is congruent to ∆ABC using the SAS Congruence Theorem (Theorem 5.5).

Question 23.

Answer:

Question 24.

Answer:

Question 25.
ERROR ANALYSIS
Describe and correct the error in finding the value of x.

Answer:

Question 26.
HOW DO YOU SEE IT?
What additional information do you need to prove that ∆ABC ≅ ∆DBC?

Answer:

Question 27.
PROOF
The Navajo rug is made of isosceles triangles. You know ∠B ≅∠D. Use the SAS Congruence Theorem (Theorem 5.5 to show that ∆ABC ≅ ∆CDE. (See Example 3.)

Answer:

Question 28.
THOUGHT PROVOKING
There are six possible subsets of three sides or angles of a triangle: SSS, SAS, SSA, AAA, ASA, and AAS. Which of these correspond to congruence theorems? For those that do not, give a counterexample.
Answer:

Question 29.
MATHEMATICAL CONNECTIONS
Prove that
∆ABC ≅ ∆DEC
Then find the values of x and y.

Answer:

Question 30.
MAKING AN ARGUMENT
Your friend claims it is possible to Construct a triangle congruent to ∆ABC by first constructing \(\overline{A B}\) and \(\overline{A C}\), and then copying ∠C. Is your friend correct? Explain your reasoning.

Answer:

Question 31.
PROVING A THEOREM
Prove the Reflections in Intersecting Lines Theorem (Theorem 4.3).
Answer:

Maintaining Mathematical Proficiency

Classify the triangle by its sides and by measuring its angles.

Question 32.

Answer:

Question 33.

Answer:

Question 34.

Answer:

Question 35.

Answer:

5.4 Equilateral and Isosceles Triangles

Exploration 1

Writing a Conjecture about Isosceles Triangles

Work with a partner: Use dynamic geometry software.

a. Construct a circle with a radius of 3 units centered at the origin.
Answer:

b. Construct ∆ABC so that B and C are on the circle and A is at the origin.
Answer:

c. Recall that a triangle is isosceles if it has at least two congruent sides. Explain why ∆ABC is an isosceles triangle.
Answer:

d. What do you observe about the angles of ∆ABC?
Answer:

e. Repeat parts (a)-(d) with several other isosceles triangles using circles of different radii. Keep track of your observations by copying and completing the table below. Then write a conjecture about the angle measures of an isosceles triangle.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures.
Answer:

f. Write the converse of the conjecture you wrote in part (e). Is the converse true?
Answer:

Communicate Your Answer

Question 2.
What conjectures can you make about the side lengths and angle measures of an
isosceles triangle?
Answer:

Question 3.
How would you prove your conclusion in Exploration 1 (e)? in Exploration 1(f)?
Answer:

Lesson 5.4 Equilateral and Isosceles Triangles

Monitoring Progress

Copy and complete the statement.

Question 1.
If \(\overline{H G}\) ≅ \(\overline{H K}\), then ∠ _______ ≅ ∠ _______ .
Answer:

Question 2.
If ∠KHJ ≅∠KJH, then ______ ≅ ______ .
Answer:

Question 3.
Find the length of \(\overline{S T}\) of the triangle at the left.
Answer:

Question 4.
Find the value of x and y in the diagram.

Answer:

Question 5.
In Example 4, show that ∆PTS ≅ ∆QTR
Answer:

Exercise 5.4 Equilateral and Isosceles Triangles

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Describe how to identify the vertex angle of an isosceles triangle.
Answer:

Question 2.
WRITING
What is the relationship between the base angles of an isosceles triangle? Explain.
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6. copy and complete the statement. State which theorem you used.

Question 3.
If \(\overline{A E}\) ≅ \(\overline{D E}\) then ∠_____ ≅ ∠_____ .
Answer:

Question 4.
If \(\overline{A B}\) ≅ \(\overline{E B}\) then ∠_____ ≅ ∠_____ .
Answer:

Question 5.
If ∠D ≅ ∠CED, then _______ ≅ _______ .
Answer:

Question 6.
If ∠EBC ≅ ∠ECB, then _______ ≅ _______ .
Answer:

In Exercises 7-10. find the value of x.

Question 7.

Answer:

Question 8.

Answer:

Question 9.

Answer:

Question 10.

Answer:

Question 11.
MODELING WITH MATHEMATICS
The dimensions of a sports pennant are given in the diagram. Find the values of x and y.

Answer:

Question 12.
MODELING WITH MATHEMATICS
A logo in an advertisement is an equilateral triangle with a side length of 7 centimeters. Sketch the logo and give the measure of each side.
Answer:

In Exercises 13-16, find the values of x and y.

Question 13.

Answer:

Question 14.

Answer:

Question 15.

Answer:

Question 16.

Answer:

CONSTRUCTION
In Exercises 17 and 18, construct an equilateral triangle whose sides are the given length.

Question 17.
3 inches
Answer:

Question 18.
1.25 inches
Answer:

Question 19.
ERROR ANALYSIS
Describe and correct the error in finding the length of \(\overline{B C}\).

Answer:

Question 20.
PROBLEM SOLVING
The diagram represents part of the exterior of the Bow Tower in Calgary. Alberta, Canada, In the diagram. ∆ABD and ∆CBD arc congruent equilateral triangles.

a. Explain why ∆ABC is isosceles.
Answer:

b. Explain ∠BAE ≅ ∠BCE.
Answer:

c. Show that ∆ABE and ∆CBE arc congruent.
Answer:

d. Find the measure of ∠BAE.
Answer:

Question 21.
FINDING A PATTERN
In the pattern shown. each small triangle is an equilateral triangle with an area of 1 square unit.

a. Explain how you know that an triangle made out of equilateral triangles is equilateral.
b. Find the areas of the first four triangles in the pattern.
c. Describe any patterns in the areas. Predict the area of the seventh triangle in the pattern. Explain your reasoning.
Answer:

Question 22.
REASONING
The base of isosceles ∆XYZ is \(\overline{Y Z}\). What
can you prove? Select all that apply.
(A) \(\overline{X Y}\) ≅ \(\overline{X Z}\)
(B) ∠X ≅ ∠Y
(C) ∠Y ≅ ∠Z
(D) \(\overline{Y Z}\) ≅ \(\overline{Z X}\)
Answer:

In Exercises 23 and 24, find the perimeter of the triangle.

Question 23.

Answer:

Question 24.

Answer:

MODELING WITH MATHEMATICS
In Exercises 25 – 28. use the diagram based on the color wheel. The 12 triangles in the diagram are isosceles triangles with congruent vertex angles.

Question 25.
Complementary colors lie directly opposite each other on the color wheel. Explain how you know that the yellow triangle is congruent to the purple triangle.
Answer:

Question 26.
The measure of the vertex angle of the yellow triangle is 30°. Find the measures of the base angles.
Answer:

Question 27.
Trace the color wheel. Then form a triangle whose vertices are the midpoints of the bases of the red. yellow. and blue triangles. (These colors are the primary colors.) What type of triangle is this?
Answer:

Question 28.
Other triangles can be brined on the color wheel that are congruent to the triangle in Exercise 27. The colors on the vertices of these triangles are called triads. What are the possible triads?
Answer:

Question 29.
CRITICAL THINKING
Are isosceles triangles always acute triangles? Explain your reasoning.
Answer:

Question 30.
CRITICAL THINKING
Is it possible for an equilateral triangle to have an angle measure other than 60°? Explain your reasoning.
Answer:

Question 31.
MATHEMATICAL CONNECTIONS
The lengths of the sides of a triangle are 3t, 5t – 12, and t + 20. Find the values of t that make the triangle isosceles. Explain your reasoning.
Answer:

Question 32.
MATHEMATICAL CONNECTIONS
The measure of an exterior angle of an isosceles triangle is x°. Write expressions representing the possible angle measures of the triangle in terms of x.
Answer:

Question 33.
WRITING
Explain why the measure of the vertex angle of an isosceles triangle must be an even number of degrees when the measures of all the angles of the triangle are whole numbers.
Answer:

Question 34.
PROBLEM SOLVING
The triangular faces of the peaks on a roof arc congruent isosceles triangles with vertex angles U and V.

a. Name two angles congruent to ∠WUX. Explain your reasoning.
b. Find the distance between points U and V.
Answer:

Question 35.
PROBLEM SOLVING
A boat is traveling parallel to the shore along \(\vec{R}\)T. When the boat is at point R, the captain measures the angle to the lighthouse as 35°. After the boat has traveled 2.1 miles, the captain measures the angle to the lighthouse to be 70°.

a. Find SL. Explain your reasoning.
b. Explain how to find the distance between the boat and the shoreline.
Answer:

Question 36.
THOUGHT PROVOKING
The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, do all equiangular triangles have the same angle measures? Justify your answer.
Answer:

Question 37.
PROVING A COROLLARY
Prove that the Corollary to the Base Angles Theorem (Corollary 5.2) follows from the Base Angles Theorem (Theorem 5.6).
Answer:

Question 38.
HOW DO YOU SEE IT?
You are designing fabric purses to sell at the school fair.

a. Explain why ∆ABE ≅ ∆DCE.
b. Name the isosceles triangles in the purse.
c. Name three angles that are congruent to ∠EAD.
Answer:

Question 39.
PROVING A COROLLARY
Prove that the Corollary to the Converse of the Base Angles Theorem (Corollary 5.3) follows from the Converse of the Base Angles Theorem (Theorem 5.7)
Answer:

Question 40.
MAKING AN ARGUMENT
The coordinates of two points are T(0, 6) and U(6, 0) Your friend claims that points T, U, and V will always be the vertices of an isosceles triangle when V is any point on the line y = x. Is your friend correct? Explain your reasoning.
Answer:

Question 41.
PROOF
Use the diagram to prove that ∆DEF is equilateral.

Given ∆ABC is equilateral
∠CAD ≅ ∠ABE ≅ ∠BCF
Prove ∆DEF is equilateral
Answer:

Maintaining Mathematical Proficiency

Use the given property to complete the statement.

Question 42.
Reflexive Property of Congruence (Theorem 2. 1): ________ ≅ \(\overline{S E}\)
Answer:

Question 43.
Symmetric Property of Congruence (Theorem 2.1): If ________ ≅ ________, then \(\overline{R S}\) ≅ \(\overline{J K}\)
Answer:

Question 44.
Transitive Property of Congruence (Theorem 2.1): If \(\overline{E F}\) ≅ \(\overline{P Q}\), and \(\overline{P Q}\) ≅ \(\overline{U V}\) ________ ≅ ________.
Answer:

5.1 to 5.4 Quiz

Find the measure of the exterior angle.

Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Identify all pairs of congruent corresponding parts. Then write another congruence statement for the polygons.

Question 4.
∆ABC ≅ ∆DEF

Answer:

Question 5.
QRST ≅ WXYZ

Answer:

Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem (Thm 5.5). If so, write a proof. If not, explain why.

Question 6.
∆CAD, ∆CBD

Answer:

Question 7.
∆GHF, ∆KHJ

Answer:

Question 8.
∆LWP, ∆NMP

Answer:

Copy and complete the statement. State which theorem you used.

Question 9.
If VW ≅ WX, then ∠______ ≅ ∠ ________.
Answer:

Question 10.
If XZ ≅ XY. then∠______ ≅ ∠ ________.
Answer:

Question 11.
If ∠ZVX ≅∠ZXV, then ∠______ ≅ ∠ ________.
Answer:

Question 12.
If ∠XYZ ≅∠ZXY, then ∠______ ≅ ∠ ________.
Answer:

Find the values of x and y.

Question 13.
∆DEF ≅ ∆QRS

Answer:

Question 14.

Answer:

Question 15.
In a right triangle, the measure of one acute angle is 4 times the difference of the measure of the other acute angle and 5. Find the measure ol each acute angle in the triangle. (Section 5.1)
Answer:

Question 16.
The figure shows a stained glass window. (Section 5.1 and Section 5.3)

a. Classify triangles 1 – 4 by their angles.
Answer:

b. Classify triangles 4 – 6 by their sides.
Answer:

c. Is there enough information given to prove that ∆7 ≅ ∆8? If so, label the vertices
and write a proof. If not, determine what additional information is needed.
Answer:

5.5 Proving Triangle Congruence by SSS

Exploration 1

Drawing Triangles

Work with a partner.
Use dynamic geometry software.

a. Construct circles with radii of 2 units and 3 units centered at the origin. Label the origin A. Then draw \(\overline{B C}\) of length 4 units.
Answer:

b. Move \(\overline{B C}\) so that B is on the smaller circle and C is on the larger circle. Then draw ∆ABC.
Answer:

c. Explain why the side lengths of ∆ABC are 2, 3, and 4 units.
Answer:

d. Find m∠A, m∠B, and m∠C.
Answer:

e. Repeat parts (b)and (d) several times, moving \(\overline{B C}\) to different locations. Keep track of ‘our results by copying and completing the table below. What can you conclude?
USING TOOLS STRATEGICALLY
To be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data.
Answer:

Communicate Your Answer

Question 2.
What can you conclude about two triangles when you know the corresponding sides are congruent?
Answer:

Question 3.
How would you prove your conclusion in Exploration 1(e)?
Answer:

Lesson 5.5 Proving Triangle Congruence by SSS

Monitoring Progress

Decide whether the congruence statement is true. Explain your reasoning.

Question 1.
∆DFG ≅ ∆HJK

Answer:

Question 2.
∆ACB ≅ ∆CAD

Answer:

Question 3.
∆QPT ≅ ∆RST

Answer:

Determine whether the figure is stable. Explain your reasoning.

Question 4.

Answer:

Question 5.

Answer:

Question 6.

Answer:

Use the diagram.

Question 7.
Redraw ∆ABC and ∆DCB side by side with corresponding parts in the same position.
Answer:

Question 8.
Use the information in the diagram to prove that ∆ABC ≅ ∆DCB.
Answer:

Exercise 5.5 Proving Triangle Congruence by SSS

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
The side opposite the right angle is called the __________of the right triangle.
Answer:

Question 2.
WHICH ONE DOESNT BELONG?
Which triangles legs do not belong with the other three? Explain your reasoning.

Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4, decide whether enough information is given to prove that the triangles are congruent using the SSS Congruence Theorem (Theorem 5.8). Explain.

Question 3.
∆ABC, ∆DBE

Answer:

Question 4.
∆PQS, ∆RQS

Answer:

In Exercises 5 and 6, decide whether enough information is given to prove that the triangles are congruent using the HL Congruence Theorem (Theorem 5.9). Explain.

Question 5.
∆ABC, ∆FED

Answer:

Question 6.
∆PQT, ∆SRT

Answer:

In Exercises 7-10. decide whether the congruence statement is true. Explain your reasoning.

Question 7.
∆RST ≅ ∆TQP

Answer:

Question 8.
∆ABD ≅ ∆CDB

Answer:

Question 9.
∆DEF ≅ ∆DGF

Answer:

Question 10.
∆JKL ≅ ∆LJM

Answer:

In Exercises 11 and 12, determine whether the figure is stable. Explain your reasoning.

Question 11.

Answer:

Question 12.

Answer:

In Exercises 13 and 14, redraw the triangles so they are side by side with corresponding parts in the same position. Then write a proof.

Question 13.
Given \(\overline{A C}\) ≅ \(\overline{B D}\)
\(\overline{A B}\) ⊥ \(\overline{A D}\)
\(\overline{C D}\) ⊥ \(\overline{A D}\)
Prove ∆BAD ≅ ∆CDA

Answer:

Question 14.
Given G is the midpoint of \(\overline{E H}\), \(\overline{F G}\) ≅ \(\overline{G I}\), ∠E and ∠H are right angles.
Prove ∆EFG ≅ ∆HIG

Answer:

In Exercises 15 and 16. write a proof.

Question 15.
Given \(\overline{L M}\) ≅ \(\overline{J K}\), \(\overline{M J}\) ≅ \(\overline{K L}\)
Prove ∆LMJ ≅ ∆JKL

Answer:

Question 16.
Given \(\overline{W X}\) ≅ \(\overline{V Z}\), \(\overline{W Y}\) ≅ \(\overline{V Y}\), \(\overline{Y Z}\) ≅ \(\overline{Y X}\)
Prove ∆VWX ≅ ∆WVZ

Answer:

CONSTRUCTION
In Exercises 17 and 18, construct a triangle that is congruent to ∆QRS using the SSS Congruence Theorem Theorem 5.8).

Question 17.

Answer:

Question 18.

Answer:

Question 19.
ERROR ANALYSIS
Describe and correct the error in identifying congruent triangles.

Answer:

Question 20.
ERROR ANALYSIS
Describe and correct the error in determining the value of x that makes the triangles congruent.

Answer:

Question 21.
MAKING AN ARGUMENT
Your friend claims that in order to use the SSS Congruence Theorem (Theorem 5.8) Lo prove that two triangles are congruent, both triangles must be equilateral triangles. Is your friend correct? Explain your reasoning.
Answer:

Question 22.
MODELING WITH MATHEMATICS
The distances between consecutive bases on a softball field are the same. The distance from home plate to second base is the same as the distance from first base to third base. The angles created at each base are 90°. Prove
∆HFS ≅ ∆FST ≅ ∆STH

Answer:

Question 23.
REASONING
To support a tree you attach wires from the trunk of the tree to stakes in the ground, as shown in the diagram.

a. What additional information do you need to use the HL Congruence Theorem (Theorem 5.9) to prove that ∆JKL ≅ ∆MKL?
b. Suppose K is the midpoint of JM. Name a theorem you could use to prove that ∆JKL ≅ ∆MKL. Explain your reasoning.
Answer:

Question 24.
REASONING
Use the photo of the Navajo rug, where \(\overline{B C}\) ≅ \(\overline{D E}\) and \(\overline{A C}\) ≅ \(\overline{C E}\)

a. What additional intormation do you need to use the SSS Congruence Theorem (Theorem 5.8) to prove that ∆ABC ≅ ∆CDE?
b. What additional information do you need to use the HL Congruence Theorem (Theorem 5.9) to prove that ∆ABC ≅ ∆CDE?
Answer:

In Exercises 25-28. use the given coordinates to determine whether ∆ABC ≅ ∆DEF.

Question 25.
A(- 2, – 2), B(4, – 2), C(4, 6), D(5, 7), E(5, 1), F(13, 1)
Answer:

Question 26.
A(- 2, 1), B(3, – 3), C(7, 5), D(3, 6), E(S, 2), F( 10, 11)
Answer:

Question 27.
A(0, 0), B(6, 5), C(9, 0), D(0, – 1), E(6, – 6), F(9, – 1)
Answer:

Question 28.
A(- 5, 7), B(- 5, 2), C(0, 2), D(0, 6), E(o, 1), F(4, 1)
Answer:

Question 29.
CRITICAL THINKING
You notice two triangles in the tile floor of a hotel lobby. You want to determine whether the triangles are congruent. but you only have a piece of string. Can you determine whether the triangles are congruent? Explain.
Answer:

Question 30.
HOW DO YOU SEE IT?
There are several theorems you can use to show that the triangles in the “square” pattern are congruent. Name two of them.

Answer:

Question 31.
MAKING AN ARGUMENT
Your cousin says that ∆JKL is congruent to ∆LMJ by the SSS Congruence Theorem (Thm. 5.8). Your friend says that ∆JKL is congruent to ∆LMJ by the HL Congruence Theorem (Thm. 5.9). Who is correct? Explain your reasoning.

Answer:

Question 32.
THOUGHT PROVOKING
The postulates and theorems in this book represent Euclidean geometry. In spherical geometry. all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry. do you think that two triangles are congruent if their corresponding sides are congruent? Justify your answer.
Answer:

USING TOOLS
In Exercises 33 and 34, use the given information to sketch ∆LMN and ∆STU. Mark the triangles with the given information.

Question 33.

Answer:

Question 34.

Answer:

Question 35.
CRITICAL THINKING
The diagram shows the light created by two spotlights, Both spotlights are the same distance from the stage.

Answer:
a. Show that ∆ABD ≅ ∆CBD. State which theorem or postulate you used and explain your reasoning.
b. Are all four right triangles shown in the diagram Congruent? Explain your reasoning.
Answer:

Question 36.
MATHEMATICAL CONNECTIONS
Find all values of x that make the triangles congruent. Explain.

Answer:

Maintaining Mathematical proficiency

Use the congruent triangles.

Question 37.
Name the Segment in ∆DEF that is congruent to \(\overline{A C}\).
Answer:

Question 38.
Name the segment in ∆ABC that is congruent to \(\overline{E F}\).
Answer:

Question 39.
Name the angle in ∆DEF that is congruent to ∠B.
Answer:

Question 40.
Name the angle in ∆ABC that is congruent to ∠F.
Answer:

5.6 Proving Triangle Congruence by ASA and AAS

Exploration 1

Determining Whether SSA Is Sufficient

Work with a partner.
a. Use dynamic geometry software to construct ∆ABC. Construct the triangle so that vertex B is at the origin. \(\overline{A B}\) has a length of 3 units. and \(\overline{B C}\) has a length of 2 units.
Answer:

b. Construct a circle with a radius of 2 units centered at the origin. Locate point D where the circle intersects \(\overline{A C}\). Draw \(\overline{B D}\).

Answer:

c. ∆ABC and ∆ABD have two congruent sides and a non included congruent angle.
Name them.
Answer:

d. Is ∆ABC ≅ ∆ABD? Explain your reasoning.
Answer:

e. Is SSA sufficient to determine whether two triangles are congruent? Explain your reasoning.
Answer:

Exploration 2

Determining Valid Congruence Theorems

Work with a partner. Use dynamic geometry software to determine which of the following are valid triangle congruence theorems. For those that are not valid. write a counter example. Explain your reasoning.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to recognize and use counterexamples.

Answer:

Communicate Your Answer

Question 3.
What information is sufficient to determine whether two triangles are congruent?
Answer:

Question 4.
Is it possible to show that two triangles are congruent using more than one congruence theorem? If so, give an example.
Answer:

Lesson 5.6 Proving Triangle Congruence by ASA and AAS

Monitoring Progress

Question 1.
Can the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use.

Answer:

Question 2.
In the diagram, \(\overline{A B}\) ⊥ \(\overline{A D}\), \(\overline{D E}\) ⊥ \(\overline{A D}\), and \(\overline{A C}\) ≅ \(\overline{D C}\) . Prove ∆ABC ≅ ∆DEF.

Answer:

Question 3.
In the diagram, ∠S ≅ ∠U and \(\overline{B D}\)\(\overline{B D}\) . Prove that ∆RST ≅ ∆VYT

Answer:

Lesson 5.6 Proving Triangle Congruence by ASA and AAS

Vocabulary and Core Concept Check

Question 1.
WRITING
How arc the AAS Congruence Theorem (Theorem 5. 11) and the ASA Congruence
Theorem (Theorem 5.10) similar? How are they different?
Answer:

Question 2.
WRITING
You know that a pair of triangles has two pairs of congruent corresponding angles. What other information do you need to show that the triangles are congruent?
Answer:

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, decide whether enough information is given to prove that the triangles are congruent. If so, state the theorem you would use.

Question 3.
∆ABC, ∆QRS

Answer:

Question 4.
∆ABC, ∆DBC

Answer:

Question 5.
∆XYZ, ∆JKL

Answer:

Question 6.
∆RSV, ∆UTV

Answer:

In Exercises 7 and 8, state the third congruence statement that is needed to prove that ∆FGH ≅ ∆LMN the given theorem.

Question 7.
Given \(\overline{G H}\) ≅ \(\overline{M N}\), ∠G ≅ ∠M, _______ = ________
Use the AAS Congruence Theorem (Thm. 5.11).
Answer:

Question 8.
Given \(\overline{F G}\) ≅ \(\overline{L M}\), ∠G ≅ ∠M, _______ = ________
Use the ASA Congruence Theorem (Thm. 5.10).
Answer:

In Exercises 9 – 12. decide whether you can use the given information to prove that ∆ABC ≅ ∆DEF Explain your reasoning.

Question 9.
∠A ≅ ∠G, ∠C ≅∠F, \(\overline{A C}\) ≅ \(\overline{D F}\)
Answer:

Question 10.
∠C ≅ ∠F, \(\overline{A B}\) ≅ \(\overline{D E}\), \(\overline{B C}\) ≅ \(\overline{E F}\)
Answer:

Question 11.
∠B ≅ ∠E, ∠C ≅∠F, \(\overline{A C}\) ≅ \(\overline{D E}\)
Answer:

Question 12.
∠A ≅ ∠D, ∠B ≅∠E, \(\overline{B C}\) ≅ \(\overline{E F}\)
Answer:

CONSTRUCTION
In Exercises 13 and 14, construct a triangle that is congruent to the given triangle using the ASA Congruence Theorem (Theorem 5.10). Use a compass and straightedge.

Question 13.

Answer:

Question 14.

Answer:

ERROR ANALYSIS
In Exercises 15 and 16, describe and correct the error.

Question 15.

Answer:

Question 16.

Answer:

PROOF
In Exercises 17 and 18, prove that the triangles are congruent using the ASA Congruence Theorem (Theorem 5.10).

Question 17.
Given M is the midpoint of \(\overline{N L}\).

Prove ∆NQM ≅ ∆MPL

Answer:

Question 18.
Given \(\overline{A J}\) ≅ \(\overline{K C}\) ∠BJK ≅ ∠BKJ, ∠A ≅ ∠C
Prove ∆ABK ≅ ∆CBJ

Answer:

PROOF
In Exercises 19 and 20, prove that the triangles are congruent using the AAS Congruence Theorem (Theorem 5.11).

Question 19.
Given \(\overline{V W}\) ≅ \(\overline{U W}\), ∠X ≅ ∠Z
Prove ∆XWV ≅ ∆ZWU

Answer:

Question 20.
Given ∠NKM ≅∠LMK, ∠L ≅∠N
Prove ∆NMK ≅ ∆LKM

Answer:

PROOF
In Exercises 21-23, write a paragraph proof for the theorem about right triangles.

Question 21.
Hypotenuse-Angle (HA) Congruence Theorem
If an angle and the hypotenuse of a right triangle are congruent to an angle and the hypotenuse of a second right triangle, then the triangles are congruent.
Answer:

Question 22.
Leg-Leg (LL) Congruence Theorem
If the legs of a right triangle are congruent to the legs of a second right triangle, then the triangles are congruent.
Answer:

Question 23.
Angle-Leg (AL) Congruence Theorem
If an angle and a leg of a right triangle are congruent to an angle and a leg of a second right triangle, then the triangles are Congruent.
Answer:

Question 24.
REASONING
What additional in information do you need to prove ∆JKL ≅ ∆MNL by the ASA Congruence Theorem (Theorem 5. 10)?

(A) \(\overline{K M}\) ≅ \(\overline{K J}\)
(B) \(\overline{K H}\) ≅ \(\overline{N H}\)
(C) ∠M ≅ ∠J
(D) ∠LKJ ≅ ∠LNM
Answer:

Question 25.
MATHEMATICAL CONNECTIONS
This toy contains △ABC and △DBC. Can you conclude that △ABC ≅ △DBC from the given angle measures? Explain

m∠ABC = (8x – 32)°
m∠DBC = (4y – 24)°
m∠BCA = (5x + 10)°
m∠BCD = (3y + 2)°
m∠CAB = (2x – 8)°
m∠CDB = (y – 6)°
Answer:

Question 26.
REASONING
Which of the following congruence statements are true? Select all that apply.

(A) \(\overline{B D}\) ≅ \(\overline{B D}\)
(B) ∆STV ≅ ∆XVW
(C) ∆TVS ≅ ∆VWU
(D) ∆VST ≅ ∆VUW
Answer:

Question 27.
PROVING A THEOREM
Prove the Converse of the Base Angles Theorem (Theorem 5.7). (Hint: Draw an auxiliary line inside the triangle.)
Answer:

Question 28.
MAKING AN ARGUMENT
Your friend claims to be able Lo rewrite any proof that uses the AAS Congruence Theorem (Thin. 5. 11) as a proof that uses the ASA Congruence Theorem (Thin. 5.10). Is this possible? Explain our reasoning.
Answer:

Question 29.
MODELING WITH MATHEMATICS
When a light ray from an object meets a mirror, it is reflected back to your eye. For example, in the diagram, a light ray from point C is reflected at point D and travels back to point A. The law of reflection states that the angle of incidence, ∠CDB. is congruent to the angle of reflection. ∠ADB.

a. Prove that ∆ABD is Congruent to ∆CBD.
Given ∠CBD ≅∠ABD
DB ⊥ AC
Prove ∆ABD ≅ ∆CBD
b. Verify that ∆ACD is isosceles.
c. Does moving away from the mirror have an effect on the amount of his or her reflection a person sees? Explain.
Answer:

Question 30.
HOW DO YOU SEE IT?
Name as man pairs of congruent triangles as you can from the diagram. Explain how you know that each pair of triangles is congruent.

Answer:

Question 31.
CONSTRUCTION
Construct a triangle. Show that there is no AAA congruence rule by constructing a second triangle that has the same angle measures but is not congruent.
Answer:

Question 32.
THOUGHT PROVOKING
Graph theory is a branch of mathematics that studies vertices and the way they are connected. In graph theory. two polygons are isomorphic if there is a one-to-one mapping from one polygon’s vertices to the other polygon’s vertices that preserves adjacent vertices. In graph theory, are any two triangles isomorphic? Explain your reasoning. second triangle that has the same angle measures but is not congruent.
Answer:

Question 33.
Mathematical Connections
Six statements are given about ∆TUV and ∆XYZ

a. List all combinations of three given statements that could provide enough information to prove that ∆TUV is congruent to ∆XYZ.
b. You choose three statements at random. What is the probability that the statements you choose provide enough information to prove that the triangles are congruent?
Answer:

Maintaining Mathematical proficiency

Find the coordinates of the midpoint of the line segment with the given endpoints.

Question 34.
C(1, 0) and D(5, 4)
Answer:

Question 35.
J(- 2, 3) and K(4, – 1)
Answer:

Question 36.
R(- 5, – 7) and S(2, – 4)
Answer:

Copy and angle using a compass and straightedge.

Question 37.

Answer:

Question 38.

Answer:

5.7 Using Congruent Triangles

Exploration 1

Measuring the Width of a River

Work with a partner:
The figure shows how a surveyor can measure the width of a river by making measurements on only one side of the river.

a. Study the figure. Then explain how the surveyor can find the width of the river.
Answer:

b. Write a proof to verify that the method you described in part (a) is valid.
Given ∠A is a right angle, ∠D is a right angle, \(\overline{A C}\) ≅ \(\overline{C D}\)
Answer:

c. Exchange Proofs with your partner and discuss the reasoning used.
CRITIQUING THE REASONING OF OTHERS
To be proficient in math, you need to listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Answer:

Exploration 2

Measuring the Width of a River

Work with a partner. It was reported that one of Napoleon’s offers estimated the width of a river as follows. The officer stood on the hank of the river and lowered the visor on his cap until the farthest thin visible was the edge of the bank on the other side. He then turned and rioted the point on his side that was in line with the tip of his visor and his eye. The officer then paced the distance to this point and concluded that distance was the width of the river.

a. Study the figure. Then explain how the officer concluded that the width of the river is EG.
Answer:

b. Write a proof to verify that the conclusion the officer made is correct.
Given ∠DEG is a right angle, ∠DEF is a right angle, ∠EDG ≅ ∠EDF
Answer:

c. Exchange proofs with your partner and discuss the reasoning used.
Answer:

Communicate Your Answer

Question 3.
How can you use congruent triangles to make an indirect measurement?
Answer:

Question 4.
Why do you think the types of measurements described in Explorations 1 and 2 are called indirect measurements?
Answer:

Lesson 5.7 Using Congruent Triangles

Monitoring Progress

Question 1.
Explain how you can prove that ∠A ≅ ∠C.

Answer:

Question 2.
In Example 2, does it mailer how far from point N you place a stake at point K? Explain.
Answer:

Question 3.
Write a plan to prove that ∆PTU ≅ ∆UQP.

Answer:

Question 4.
Use the construction of an angle bisector on page 42. What segments can you assume are congruent?
Answer:

Exercise 5.7 Using Congruent Triangles

Vocabulary and core concept check

Question 1.
COMPLETE THE SENTENCE
_____________ parts of congruent triangle are congruent.
Answer:

Question 2.
WRITING
Describe a situation in which you might choose to use indirect measurement with
congruent triangles to find a measure rather than measuring directly.
Answer:

Monitoring Progress and Modeling With Mathematics

In Exercise 3-8, explain how to prove that the statement is true.

Question 3.
∠A ≅ ∠D

Answer:

Question 4.
∠Q ≅∠T

Answer:

Question 5.
\(\overline{J M}\) ≅ \(\overline{L M}\)

Answer:

Question 6.
\(\overline{A C}\) ≅ \(\overline{D B}\)

Answer:

Question 7.
\(\overline{G K}\) ≅ \(\overline{H J}\)

Answer:

Question 8.
\(\overline{Q W}\) ≅ \(\overline{V T}\)

Answer:

In Exercises 9-12, write a plan to prove that ∠1 ≅∠2.

Question 9.

Answer:

Question 10.

Answer:

Question 11.

Answer:

Question 12.

Answer:

In Exercises 13 and 14. write a proof to verify that the construction is valid.

Question 13.
Line perpendicular to a line through a point not on the line

Plan for proof ∆APQ ≅ ∆BPQ by the congruence Theorem (Theorem 5.8). Then show the ∆APM ≅ ∆BPM using the SAS Congruence Theorem (Theorem 5.5). Use corresponding parts of congruent triangles to show that ∠AMP and ∠BMP are right angles.
Answer:

Question 14.
Line perpendicular to a line through a p0int on the line

Plan for Proof Show that ∆APQ ≅ ∆BPQ by the SSS Congruence Theorem (Theorem 5.8) Use corresponding parts of congruent triangles to show that ∠QPA and ∠QPB are right angles.
Answer:

In Exercises 15 and 16, use the information given in the diagram to write a proof.

Question 15.
Prove \(\overline{F L}\) ≅ \(\overline{H N}\)

Answer:

Question 16.
Prove ∆PUX ≅ ∆QSY

Answer:

Question 17.
MODELING WITH MATHEMATICS
Explain how to find the distance across the canyon.

Answer:

Question 18.
HOW DO YOU SEE IT?
Use the tangram puzzle.
Answer:

a. Which triangle(s) have an area that is twice the area of the purple triangle?
b. How man times greater is the area of the orange triangle than the area of the purple triangle?
Answer:

Question 19.
PROOF
Prove that the green triangles in the Jamaican flag congruent if \(\overline{A D}\) || \(\overline{B C}\) and E is the midpoint of \(\overline{A C}\).

Answer:

Question 20.
THOUGHT PROVOKING
The Bermuda Triangle is a region in the Atlantic Ocean in which many ships and planes have mysteriously disappeared. The vertices are Miami. San Juan. and Bermuda. Use the Internet or some other resource to find the side lengths. the perimeter, and the area of this triangle (in miles). Then create a congruent triangle on land using cities as vertices.

Answer:

Question 21.
MAKING AN ARGUMENT
Your friend claims that ∆WZY can be proven congruent to ∆YXW using the HL Congruence Theorem (Thm. 5.9). Is your friend correct? Explain your reasoning.

Answer:

Question 22.
CRITICAL THINKING
Determine whether each conditional statement is true or false. If the statement is false, rewrite it as a true statement using the converse, inverse, or contrapositive.
a. If two triangles have the same perimeter, then they are congruent.
b. If two triangles are congruent. then they have the same area.
Answer:

Question 23.
ATTENDING TO PRECISION
Which triangles are congruent to ∆ABC? Select all that apply.

Answer:

Maintaining Mathematical Proficiency

Find the perimeter of the polygon with the given vertices.

Question 24.
A(- 1, 1), B(4, 1), C(4, – 2), D(- 1, – 2)
Answer:

Question 25.
J(- 5, 3), K(- 2, 1), L(3, 4)
Answer:

5.8 Coordinate Proofs

Exploration 1

Writing a coordinate Proof

Work with a partner.

a. Use dynamic geometry software to draw \(\overline{A B}\) with endpoints A(0, 0) and B(6, 0).
Answer:

b. Draw the vertical line x = 3.
Answer:

c. Draw ∆ABC so that C lies on the line x = 3.
Answer:

d. Use your drawing to prove that ∆ABC is an isosceles triangle.
Answer:

Exploration 2

Writing a Coordinate proof

Work with a partner.

a. Use dynamic geometry software to draw \(\overline{A B}\) with endpoints A(0, 0) and B(6, 0).

b. Draw the vertical line x = 3.

c. Plot the point C(3, 3) and draw ∆ABC. Then use your drawing to prove that ∆ABC is an isosceles right triangle.

d. Change the coordinates of C so that C lies below the x-axis and ∆ABC is an isosceles right triangle.
Answer:

e. Write a coordinate proof to show that if C lies on the line x = 3 and ∆ABC is an isosceles right triangle. then C must be the point (3, 3) or the point found in part (d).
CRITIQUING THE REASONING OF OTHERS
To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.
Answer:

Communicate Your Answer

Question 3.
How can you use a coordinate plane to write a proof?
Answer:

Question 4.
Write a coordinate proof to prove that ∆ABC with vertices A(0, 0), 8(6, 0), and C(3, 3√3) is an equilateral triangle.
Answer:

Lesson 5.8 Coordinate Proofs

Monitoring Progress

Question 1.
Show another way to place the rectangle in Example 1 part (a) that is convenient
for finding side lengths. Assign new coordinates.
Answer:

Question 2.
A square has vertices (0, 0), (m, 0), and (0, m), Find the fourth vertex.
Answer:

Question 3.
Write a plan for the proof.
Given \(\vec{G}\)J bisects ∠OGH.
Proof ∆GJO ≅ ∆GJH

Answer:

Question 4.
Graph the points 0(0, 0), H(m, n), and J(m, 0). Is ∆OHJ a right triangle? Find the side lengths and the coordinates of the midpoint of each side.
Answer:

Question 5.
Write a coordinate proof.

Given Coordinates of vertices of ∆NPO and ∆NMO
Prove ∆NPO ≅ ∆NMO
Answer:

Exercise 5.8 Coordinate Proofs

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
How is a coordinate proof different from other types of proofs you have studied?
How is it the same?
Answer:

Question 2.
WRITING
Explain why it is convenient to place a right triangle on the grid as shown when writing a coordinate proof.

Answer:

Maintaining Progress and Modeling with Mathematics

In Exercises 3-6, place (he figure in a coordinate plane in a convenient way. Assign coordinates to each vertex. Explain the advantages of your placement.

Question 3.
a right triangle with leg lengths of 3 units and 2 units
Answer:

Question 4.
a square with a side length of 3 units
Answer:

Question 5.
an isosceles right triangle with leg length p
Answer:

Question 6.
a scalene triangle with one side length of 2m
Answer:

In Exercises 7 and 8, write a plan for the proof.

Question 7.
Given Coordinates of vertices of ∆OPM and ∆ONM Prove ∆OPM and ∆ONM are isosceles triangles.

Answer:

Question 8.
Given G is the midpoint of \(\overline{H F}\).
Prove ∆GHJ ≅ ∆GFO

Answer:

In Exercises 9-12, place the figure in a coordinate plane and find the indicated length.

Question 9.
a right triangle with leg lengths of 7 and 9 units; Find the length of the hypotenuse.
Answer:

Question 10.
an isosceles triangle with a base length of 60 units and a height of 50 units: Find the length of one of the legs.
Answer:

Question 11.
a rectangle with a length o! 5 units and a width of 4 units: Find the length of the diagonal.
Answer:

Question 12.
a square with side length n: Find the length of the diagonal.
Answer:

In Exercises 13 and 14, graph the triangle with the given vertices. Find the length and the slope of each side of the triangle. Then find the coordinates of the midpoint of each side. Is the triangle a right triangle? isosceles? Explain. Assume all variables are positive and in m ≠ n.)

Question 13.
A(0, 0), B(h, h), C(2h, 0)
Answer:

Question 14.
D(0, n), E(m, n), F(m, 0)
Answer:

In Exercises 15 and 16, find the coordinates of any unlabeled vertices. Then find the indicated length(s).

Question 15.
Find ON and MN.

Answer:

Question 16.
Find OT.

Answer:

PROOF
In Exercises 17 and 18, rite a coordinate proof.

Question 17.
Given Coordinates of vertices of ∆DEC and ∆BOC
Prove ∆DEC ≅ ∆BOC

Answer:

Question 18.
Given Coordinates of ∆DEA, H is the midpoint of \(\overline{D A}\), G is the mid point of \(\overline{E A}\)
Prove \(\overline{D G}\) ≅ \(\overline{E H}\)

Answer:

Question 19.
MODELING WITH MATHEMATICS
You and your cousin are camping in the woods. You hike to a point that is 500 meters cast and 1200 meters north of the Campsite. Your cousin hikes to a point that is 1000 meters cast of the campsite. Use a coordinate proof to prove that the triangle formed by your Position, your Cousin’s position. and the campsite is isosceles. (See Example 5.)

Answer:

Question 20.
MAKING AN ARGUMENT
Two friends see a drawing of quadrilateral PQRS with vertices P(0, 2), Q(3, – 4), R(1, – 5), and S(- 2, 1). One friend says the quadrilateral is a parallelogram but not a rectangle. The other friend says the quadrilateral is a rectangle. Which friend is correct? Use a coordinate proof to support your answer.
Answer:

Question 21.
MATHEMATICAL CONNECTIONS
Write an algebraic expression for the coordinates of each endpoint of a line segment whose midpoint is the origin.
Answer:

Question 22.
REASONING
The vertices of a parallelogram are (w, 0), (o, v), (- w, 0), and (0, – v). What is the midpoint of the side in Quadrant III?
(a) \(\left(\frac{w}{2}, \frac{v}{2}\right)\)
(b) \(\left(-\frac{w}{2},-\frac{v}{2}\right)\)
(c) \(\left(-\frac{w}{2}, \frac{v}{2}\right)\)
(d) \(\left(\frac{w}{2},-\frac{v}{2}\right)\)
Answer:

Question 23.
REASONING
A rectangle with a length of 3h and a width of k has a vertex at (- h, k), Which point cannot be a vertex of the rectangle?
(A) (h, k)
(B) (- h, 0)
(c) (2h, 0)
(D) (2h, k)
Answer:

Question 24.
THOUGHT PROVOKING
Choose one of the theorems you have encountered up to this point that you think would be easier to prove with a coordinate proof than with another type of proof. Explain your reasoning. Then write a coordinate proof.
Answer:

Question 25.
CRITICAL THINKING
The coordinates of a triangle are (5d – 5d), (0, – 5d), and (5d, 0). How sh
would the coordinates be changed to make a coordinate proof easier to complete?
Answer:

Question 26.
HOW DO YOU SEE IT?
without performing any calculations, how do you know that the diagonals of square TUVW are perpendicular to each oilier? How can you use a similar diagram to show that the diagonals of any square are perpendicular to each other?

Answer:

Question 27.
PROOF
Write a coordinate proof for each statement.
a. The midpoint o! the hypotenuse of a right triangle is the same distance from each vertex of the triangle.
b. Any two congruent right isosceles triangles can be combined to form a single isosceles triangle.
Answer:

Maintaining Mathematical proficiency

\(\vec{Y}\)W bisects ∠XYZ such that m∠XYW = (3x – 7)° and m∠WYZ = (2x + 1)°.

Question 28.
Find the value of x.
Answer:

Question 29.
Find m∠XYZ
Answer:

Congruent Triangles Chapter Review

5.1 Angles of Triangles

Question 1.
Classify the triangle at the right by its sides and by measuring its angles.

Answer:

Find the measure of the exterior angle.

Question 2.

Answer:

Question 3.

Answer:

Find the measure of each acute angle.

Question 4.

Answer:

Question 5.

Answer:

5.2 Congruent Polygons

Question 6.
In the diagram. GHJK ≅ LMNP. Identify all pairs of congruent corresponding parts. Then write another congruence statement for the quadrilaterals.

Answer:

Question 7.
Find m ∠ V.

Answer:

5.3 Proving Triangle Congruence by SAS

Decide whether enough information is given to prove that ∆WXZ ≅ ∆YZX using the SAS Congruence Theorem (Theorem 5.5). If so, write a proof. If not, explain why.

Question 8.

Answer:

Question 9.

Answer:

5.4 Equilateral and Isosceles Triangles

Copy and Complete the statement.

Question 10.
If \(\overline{Q P}\) ≅ \(\overline{Q R}\), then ∠ ______ ≅ ∠ ______ .
Answer:

Question 11.
If ∠TRV ≅ ∠TVR, then ______ ≅ ______ .
Answer:

Question 12.
If \(\overline{R Q} \cong \overline{R S}\), then ∠ ______ ≅ ∠ ______ .
Answer:

Question 13.
If ∠SRV ≅ ∠SVR, then ______ ≅ ______ .
Answer:

Question 14.
Find the values of x and y in the diagram.

Answer:

5.5 Proving Triangle Congruence by SSS

Question 15.
Decide whether enough information is given to prose that ∆LMP ≅ ∆NPM using the SSS Congruence Theorem (Thin. 5.8). If so, write a proof. If not, explain why.

Answer:

Question 16.
Decide whether enough information is given to prove that ∆WXZ ≅ ∆YZX using the HL Congruence Theorem (Thm. 5.9). If so, write a proof. If not, explain why.

Answer:

5.6 Proving Triangle Congruence by ASA and AAS

Question 17.
∆EFG, ∆HJK

Answer:

Question 18.
∆TUS, ∆QRS

Answer:

Decide whether enough information is given to prove that the triangles are congruent using the ASA Congruence Theorem (Thm. 5.10). If so, write a proof, If not, explain why.

Question 19.
∆LPN, ∆LMN

Answer:

Question 20.
∆WXZ, ∆YZX

Answer:

5.7 Using Congruent Triangles

Question 21.
Explain how to prove that ∠K ≅∠N.

Answer:

Question 22.
Write a plan to prkove that ∠1 ≅ ∠2

Answer:

5.8 Coordinate Proofs

Question 23.
Write a coordinate proof.
Given Coordinates of vertices of quadrilateral OPQR
Prove ∆OPQ ≅ ∆QRO

Answer:

Question 24.
Place an isosceles triangle in a coordinate plane in a way that is convenient for finding side lengths. Assign coordinates to each vertex.
Answer:

Question 25.
A rectangle has vertices (0, 0), (2k, 0), and (0, k), Find the fourth vertex.
Answer:

Congruent Triangles Test

Write a Proof.

Question 1.
Given \(\overline{C A} \cong \overline{C B} \cong \overline{C D} \cong \overline{C E}\)
Prove ∆ABC ≅ ∆EDC

Answer:

Question 2.
Given \(\overline{J K}\|\overline{M L}, \overline{M J}\| \overline{K L}\)
Prove ∆MJK ≅ ∆KLM

Answer:

Question 3.
Gven \overline{Q R} \cong \overrightarrow{R S}\(\), ∠P ≅ ∠T
Prove ∆SRP ≅ ∆QRT

Answer:

Question 4.
Find the measure of each acute angle in the figure at the right.

Answer:

Question 5.
Is it possible to draw an equilateral triangle that is not equiangular? If so, provide an example. If not, explain why.
Answer:

Question 6.
Can you use the Third Angles Theorem (Theorem 5.4) to prove that two triangles are congruent? Explain your reasoning.
Answer:

Write a plan through that ∠1 ≅∠2

Question 7.

Answer:

Question 8.

Answer:

Question 9.
Is there more than one theorem that could be used to prove that ∆ABD ≅ ∆CDB? If so, list all possible theorems.

Answer:

Question 10.
Write a coordinate proof t0 show that the triangles created b the keyboard stand are congruent.

Answer:

Question 11.
The picture shows the Pyramid of Cestius. which is located in Rome, Italy. The measure of the base for the triangle shown is 100 Roman feet. The measures of the other two sides of the triangle are both 144 Roman feet.

a. Classify the triangle shown by its sides.
Answer:

b. The measure of ∠3 is 40° What are the measures of ∠1 and ∠2? Explain your reasoning.
Answer:

Congruent Triangles Cumulative Assessment

Question 1.
Your friend claims that the Exterior Angle Theorem (Theorem 5.2) can be used to prove the Triangle Sum Theorem (Theorem 5, 1). Is your friend correct? Explain your reasoning.
Answer:

Question 2.
Use the steps in the construction to explain how you know that the line through point P is parallel to line m.

Answer:

Question 3.
The coordinate plane shows ∆JKL and ∆XYZ

a. Write a composition of transformations that maps ∆JKL to ∆XYZ
Answer:

b. Is the composition a congruence transformation? If so, identify all congruent corresponding parts.
Answer:

Question 4.
The directed line segment RS is shown. Point Q is located along \(\overline{R S}\) so that the ratio of RQ to QS is 2 to 3. What are the coordinates of point Q?

(A) Q(1, 2, 3)
(B) Q(4, 2)
(C) Q(2, 3)
(D) Q(-6, 7)
Answer:

Question 5.
The coordinate plane shows that ∆ABC and ∆DEF

a. Prove ∆ABC ≅ ∆DEF using the given information.
Answer:

b. Describe the composition of rigid motions that maps ∆ABC to ∆DEF
Answer:

Question 6.
The vertices of a quadrilateral are W(0, 0), X(- 1, 3), )(2, 7), and Z(4, 2). Your friend claims that point W will not change after dilatinig quadrilateral WXYZ by a scale factor of 2. Is your friend correct? Explain your reasoning.
Answer:

Question 7.
Which figure(s) have rotational symmetry? Select all that apply.
(A)
(B)
(C)
(D)
Answer:

Question 8.
Write a coordinate proof.
Given Coordinates of vertices of quadrilateral ABCD
Prove Quadrilateral ABCD is a rectangle.

Answer:

Question 9.
Write a proof to verify that the construction of the equilateral triangle shown below is valid.

Answer:

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Big Ideas 4th Grade Math Book Answer Key Chapter 6 Factors, Multiples, and Patterns

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Lesson: 1 Understand Factors

• Lesson 6.1 Understand Factors
• Understand Factors Homework & Practice 6.1

Lesson: 2 Factors and Divisibility

• Lesson 6.2 Factors and Divisibility
• Factors and Divisibility Homework & Practice 6.2

Lesson: 3 Relate Factors and Multiples

• Lesson 6.3 Relate Factors and Multiples
• Relate Factors and Multiples Homework & Practice 6.3

Lesson: 4 Identify Prime and Composite Numbers

• Lesson 6.4 Identify Prime and Composite Numbers
• Identify Prime and Composite Numbers Homework & Practice 6.4

Lesson: 5 Number Patterns

• Lesson 6.5 Number Patterns
• Number Patterns Numbers Homework & Practice 6.5

Lesson: 6 Shape Patterns

• Lesson 6.6 Shape Patterns
• Shape Patterns Numbers Homework & Practice 6.6

Performance Task

• Factors, Multiples, and Patterns Performance Task
• Factors, Multiples, and Patterns Activity
• Factors, Multiples, and Patterns Chapter Practice

Lesson 6.1 Understand Factors

Explore and Grow

Draw two different rectangles that each have an area of 24 square units. Label their side lengths.

Compare your rectangles to your partner’s rectangles. How are they the same? How are they different?

Answer:

Explanation:
My rectangle is 6 × 4 and my partner’s rectangle is 3 × 8
We know that,
Area of rectangle = l × b
Hence,
According to the above formula, the areas of rectangles are the same irrespective of different lengths and different breadths.

Structure
How is each side length related to 24?

Answer: 
We know that,
The factors of 24 = 4 × 6, 3 × 8, 1 × 24, 2 × 12
Remember that,
a × b = b × a
So,
4 × 6 = 6 × 4
This pattern will be applicable to all factor pairs
Hence, from the above,
We can conclude that each side length is related to 24 because of the factor pair.

Think and Grow: Find Factor Pairs

You can write whole numbers as products of two factors. The two factors are called a factor pair for the number

Example
Find the factor pairs for 20.
Find the side lengths of as many different rectangles with an area of 20 square units as possible.

The side lengths of each rectangle are a factor pair.
So, the factor pairs for 20 are 12 and 1,  10 and 2,  and 5 and 4.

Show and Grow

Question 1.
Use the rectangles to find the factor pairs for 12.

Answer: Factor pairs of 12 are: 12 and 1, 6 and 2, and 4 and 3

Explanation:

The factor pairs are nothing but the side lengths of a rectangle and the area of a rectangle gives the factor
Hence,
The factor pairs of 12 are: 12 and 1, 6 and 2, and 4 and 3

Question 2.
Draw rectangles to find the factor pairs for 16.

Answer: The factor pairs of 16 are: 1 and 16, 2 and 8, 4 and  4

Explanation:

The factor pairs are nothing but the side lengths of a rectangle and the area of a rectangle gives the factor
Hence,
The factor pairs of 16 are: 1 and 16, 2 and 8, 4 and  4

Apply and Grow: Practice

Draw rectangles to find the factor pairs for the number.
Question 3.
14

Answer: The factor pairs of 14 are: 1 and 14, 2 and 7

Explanation:

Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 14 are: 1 and 14, 2 and 7

Question 4.
15

Answer:  The factor pairs of 15 are: 1 and 15, 3 and 5

Explanation:

Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 15 are: 1 and 15, 3 and 5

Question 5.
20

Answer: The factor pairs of 20 are: 1 and 20, 2 and 10, 4 and 5

Explanation:

Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 15 are: 1 and 15, 3 and 5

Question 6.
36

Answer: The factor pairs of 36 are: 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6

Explanation:

Factors are the numbers that divide the number completely.
Hence,
The factor pairs of 36 are: 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6

Find the factor pairs for the number.
Question 7.
11
Answer: The factor pairs of 11 are: 1 and 11

Explanation:

Factors are the numbers that divide the original number completely.Hence,
The factor pairs of 15 are: 1 and 15, 3 and 5

Question 8.
9
Answer: The factor pairs of 9 are: 1 and 9, and 3 and 3

Explanation:

Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 15 are: 1 and 15, 3 and 5

Question 9.
4

Answer: The factor pairs of 4 are: 1 and 4, 2 and 2

Explanation:
Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 4 are:
1 × 4, 2 × 2 ( Since 4 ×1 and 1 × 4 are equal, we will take any 1 factor )

Question 10.
25
Answer:  The factor pairs of 25 are: 1 and 25, 5 and 5

Explanation:
Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 25 are:
1 × 25, 5 × 5

Question 11.
10
Answer: The factor pairs of 10 are: 1 and 10, 2 and 5

Explanation:
Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 10 are:
1 × 10, 5 × 5

Question 12.
40
Answer: The factor pairs of 40 are: 1 and 40, 2 and 20, 4 and 10, 5 and 8

Explanation:
Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 40 are:
1 × 40, 2 × 20, 4 × 10 and 5 × 8

Question 13.
Writing
Use the word to explain one way that 2 and 6 are related.
Answer: The factor pair of 6 are: 1 and 6, 2 and 3

Explanation:
The one way 2 and  are related is the “Factor-pair method”
According to the factor-pair method,
The factors of 6 are:
1 × 6 and 2 × 3
Hence, from the above,
We can conclude that 2 and 6 are related due to the factor-pair method.

Think and Grow: Modeling Real Life

Example
You want to organize 30 pictures into a rectangular array on a wall. How many different arrays can you make?

To find the number of arrays you can make, find the number of factor pairs for 30.
There are 3-factor pairs for 30.
You can use each factor pair to make 30 arrays.
So, there are 30 ×3 = 90 ways to organize the pictures in different arrays.

Show and Grow

Question 14.
A city mayor buys 27 solar panels. She wants to organize the panels into a rectangular array. How many different arrays can she make?

Answer: The different arrays she can make are: 1 × 27, 3 × 9, 9 × 3, and 27 × 1

Explanation:
Given that a city mayor buys 27 solar panels and she wants to organize the panels into a rectangular array.
The “Array” is nothing but the number of patterns ( Factor pairs)  that we can arrange the given things.
Hence,
The factors of 27 are: 1 and 27, 3 and 9, 9 and 3, and 27 and 1
Hence,
The different arrays she can make in arranging the solar panels are:
1 × 27, 3 × 9, 9 × 3, and 27 × 1

Question 15.
DIG DEEPER!
A store owner has 42 masks to hang in a rectangular array on a wall. The owner does not have room for more than 10 masks in each row or column. What are the possible numbers of masks the owner should hang in each row?

Answer: The number of possible masks the owner should hang in each row is: 7 masks

Explanation:
Given that a store owner has 42 masks to hang in a rectangular array on a wall.
So now,
The number of arrays ( Factor pairs ) of 42 are:
1 × 42, 2 × 21, 3 × 14, 6 × 7, 7 × 6, 14 × 3, 21 × 2, and 41 × 1
It is also given that the owner does not have enough room for more than 10 masks in each row or column.
So, from the above factor pairs, we can say that the possible array of masks is: 6 × 7 and 7 × 6
Hence, from the above,
we can conclude that the maximum number of masks the owner can put either in rows or columns is: 7 masks

Question 16.
A teacher wants to set up a chair for each of the 48 students in chorus. He wants to set up the chairs in a rectangular array. He can fit no more than 20 rows and no more than 30 chairs in each row in the room. What are the possible numbers of rows that he could set up?

Answer:  The possible number of rows that he could set up are: 16 rows

Explanation:
Given that a teacher wants to set up a chai for each of the 48 students in the chorus and he wants to set up the chairs in a rectangular array.
Hence, the number of arrays ( factor pairs) of 48 are:
1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8, 8 × 6, 12 × 4, 16 × 3, 24 × 2, and 48 × 1
It is also given that he can fit no more than 20 rows and no more than 30 chairs in each row in the room.
Hence, from the above,
We can conclude that the possible number of rows that he could set up are: 16 rows

Understand Factors Homework & Practice 6.1

Question 1.
Use the rectangles to find the factor pairs for 8.

Answer: The factors of 8 are: 1 and 8, 2 and 4

Explanation:

Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 8 are: 1 × 8 and 2 × 4

Question 2.
Draw rectangles to find the factor pairs for 21.

Answer: The factor pairs of 21 are: 1 × 21, 3 × 7

Explanation:

Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 21 are: 1 × 21, 3 × 7

Question 3.
Draw rectangles to find the factor pairs for 28.

Answer: The factor pairs of 28 are: 1 × 28, 2 × 14, 4 × 7

Explanation:


Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 28 are: 1 × 28, 2 × 14, 4 × 7

Find the factor pairs for the number.
Question 4.
13
Answer: The factor pairs of 13 are: 1 × 13 and 13 × 1

Explanation:
Factors are the numbers that divide the original number completely. Hence,
The factor pairs of 13 are:
1 × 13 and 13 × 1

Question 5.
5
Answer: The factor pairs of 5 are: 1 × 5 and 5 × 1

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 5 are:
1 × 5 and 5 × 1

Question 6.
35
Answer: The factor pairs of 35 are: 1 × 35, 5 × 7

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 35 are:
1 × 35, 5 × 7

Question 7.
45
Answer: The factor pairs of 45 are: 1 × 45, 3 × 15, 5× 9

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 45 are:
1 × 45, 3 × 15, 5× 9

Question 8.
18
Answer: The factor pairs of 18 are: 1 × 18, 2 × 9, 3 × 6

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 18 are:
1 × 18, 2 × 9, 3 × 6

Question 9.
36
Answer: The factor pairs of 36 are: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 36 are:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6

Question 10.
YOU BE THE TEACHER
Descartes says there are 5-factor pairs for 16. Newton says there are 3-factor pairs for 16. Who is correct? Explain.

Answer:  Newton is correct

Explanation:
We know that,
a × b = b × a
Hence,
The factor pairs of 16 are:
1 × 16, 2 × 8, 4 × 4, 8 × 2, 16 × 1
But according to the above,
We don’t have to consider the factor pairs 8 × 2 and 16 × 1
Hence,
There are only 3-factor pairs.
But, Descartes says that there are 5-factor pairs for 16 while Newton says it three.
Hence, from the above,
We can conclude that Newton is correct.

Question 11.
Modeling Real Life
A race volunteer has 50 cases of bottled water. He wants to arrange the cases into a rectangular array. How many different arrays can he make?

Answer: The number of different arrays he can make is: 6 arrays

Explanation:
Given that a race volunteer has 50 cases of bottled water and he wants to arrange the cases into a rectangular array.
The array is nothing but the factor pairs.
Hence, the number of arrays of 50 he can make = 1 × 50, 2 × 25, 5 × 10, 10 × 5, 25 × 2, 50 × 1
Hence, from the above,
we can conclude that the number of different arrays to arrange the 50 cases of bottled water are:
a) 1 × 50 b)  2 × 25 c)  5 × 10 d)  10 × 5 e)  25 × 2 f)  50 × 1

Review & Refresh

Find the product.
Question 12.
2 × 14 = _____
Answer: 2 × 14 = 28

Explanation:
According to the Distributive Property of Multiplication,
2 × 14 = ( 10 + 4 ) × 2
= ( 2 × 10 ) + ( 2 × 4 )
= 20 + 8
= 28
Hence, 2 × 14 = 28

Question 13.
22 × 7 = ______
Answer: 22 × 7 = 154

Explanation:
According to the Distributive Property of Multiplication,
22 × 7 = ( 20 + 2 ) × 7
= ( 20 × 7 ) + ( 2 × 7 )
= 140 + 14
= 154
Hence, 22 × 7 = 154

Question 14.
9 × 27 = ______
Answer: 27 × 9 = 243

Explanation:
According to the Distributive Property of Multiplication,
27 × 9 = ( 20 + 7 ) × 9
= ( 20 × 9 ) + ( 7 × 9 )
= 180 + 63
= 243
Hence, 27 × 9 = 243

Lesson 6.2 Factors and Divisibility

Explore and Grow

List any 10 multiples of 3. What do you notice about the sum of the digits in each multiple?
Answer:
The 10 multiples of 3 are:
3, 6, 9, 12, 15, 18 ,21, 24, 27, and 30
From the sum of the digits in the above multiples,
0 + 3 =3
0 + 6 = 6
0 + 9 = 9
1 + 2 = 3
1 + 5 = 6
2 + 1 = 3
2 + 4= 6
2 + 7 = 9
3 + 0 = 3
Hence, from the above sums,
We can conclude that the sum of the digits in the 10 multiples are also the multiples of 3.

List any 10 multiples of 9. What do you notice about the sum of the digits in each multiple?
Answer:
The 10 multiples of 9 are:
9, 18, 27, 36, 45, 54, 63, 72, 81, and 90
From the sum of the digits in the above multiples,
0 + 9 = 9
1 + 8 = 9
2 + 7 = 9
3 + 6 = 9
4 + 5 = 9
5 + 4 = 9
6 + 3 = 9
7 + 2 = 9
8 + 1 = 9
9 + 0 = 9
Hence, from the above,
We can conclude that the sum of the digits in all the multiples of 9 is equal to 9 only.

Structure
How can you use your observations above to determine whether 3 and 9 are factors of a given number? Explain.
Answer: The factors of 9 are: 1,3, and 9

Explanation:
The factor pairs of 9 are:
1 × 9, 3 × 3 and 9 × 1
Hence, from the above,
We can conclude that 3 and 9 are the factors of 9.

Think and Grow: Find Factors and Factor Pairs

A number is divisible by another number when a quotient is a whole number and the remainder is 0.

Some numbers have divisibility rules that you can use to determine whether they are factors of other numbers.

Example
Find the factor pairs for 48.
Use divisibility rules and division to find the factors of 48.

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
The factor pairs for 48 are 6.

Show and Grow

Find the factor pairs for the number.
Question 1.
30
Answer:
The factor pairs of 30 are:
1 × 30, 2 × 15, 3 × 10, 5 × 6, 6 × 5, 10 × 3, 15 × 2, 30 × 1

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 30 are:
1 × 30, 2 × 15, 3 × 10, 5 × 6, 6 × 5, 10 × 3, 15 × 2, 30 × 1

Question 2.
54
Answer:
The factor pairs of 54 are:
1 × 54, 2 × 27, 3 × 18, 6 × 9, 9 × 6, 18 × 3, 27 × 2, 54 × 1

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 54 are:
1 × 54, 2 × 27, 3 × 18, 6 × 9, 9 × 6, 18 × 3, 27 × 2, 54 × 1

Apply and Grow: Practice

Find the factor pairs for the number.
Question 3.
29
Answer:
The factor pairs of 29 are:
1 × 29 and 29 × 1

Explanation:
Factors are the numbers that divide the original numbers completely.
Hence,
The factor pairs of 29 are:
1 × 29 and 29 × 1

Question 4.
50
Answer:
The factor pairs of 50 are:
1 × 50, 2 × 25, 5 × 10, 10 × 5, 25 × 2, and 50 × 1

Explanation:
Factors are the numbers that divide the original numbers completely.
Hence,
The factor pairs of 50 are:
1 × 50, 2 × 25, 5 × 10, 10 × 5, 25 × 2, and 50 × 1

Question 5.
63
Answer:
The factor pairs of 63 are:
1 × 63, 3 × 21, 7 × 9, 9 × 7, 21 × 3 and 63 × 1

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factor pairs of 63 are:
1 × 63, 3 × 21, 7 × 9, 9 × 7, 21 × 3 and 63 × 1

Question 6.
33
Answer:
The factor pairs of 33 are:
1 × 33, 3 × 11, 11 × 3 and 33 × 1

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factor pairs of 33 are:
1 × 33, 3 × 11, 11 × 3 and 33 × 1

Question 7.
60
Answer:
The factor pairs of 60 are:
1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, and 6 × 10

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factor pairs of 60 are:
1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, and 6 × 10

Question 8.
64
Answer:
The factor pairs of 64 are:
1 × 64, 2 × 32, 4 × 16, and 8 ×8

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factor pairs of 64 are:
1 × 64, 2 × 32, 4 × 16, and 8 ×8

List the factors of the number.
Question 9.
39
Answer:
The factors of 39 are: 1, 3, 13, and 39

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factors of 39 are: 1, 3, 13, and 39

Question 10.
44
Answer:
The factors of 44 are: 1, 2, 4, 11, 22, and 44

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factors of 44 are: 1, 2, 4, 11, 22, and 44

Question 11.
72
Answer:
The factors of 72 are: 1, 2, 3,4, 6, 8, 9, 12, 18, 24, 36, and 72

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factors of 72 are: 1, 2, 3,4, 6, 8, 9, 12, 18, 24, 36, and 72

Question 12.
67
Answer:
The factors of 7 are: 1 and 67

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factors of 7 are: 1 and 67

Question 13.
42
Answer:
The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, and 42

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, and 42

Question 14.
28
Answer:
The factors of 28 are:
1, 2, 4, 7, 14, and 28

Explanation:
Factors are the numbers that divide the number originally
Hence,
The factors of 28 are:
1, 2, 4, 7, 14, and 28

Question 15.
Reasoning
Can an odd number have an even factor? Explain.
Answer: No, an odd number doesn’t have an even factor

Explanation:
Let the number which we want to find the factors is: 15
The factors of 15 are:
1, 3, 5, and 15
Hence, from the above factors,
We can conclude that an odd number doesn’t have an even factor.
Note: The “Odd number” is the number that can’t be divisible by 2.

Question 16.
Writing
Use the diagram to explain why you do not have to check whether any numbers greater than 4 are factors of 12.

Answer: We don’t have to check the numbers greater than 4 are the factors of 12 because the factors of 12 greater than 4 are repeating.

Explanation:
The given number is 12
The factors of 12 are:
1, 2, 3, 4, 6, and 12
Hence, from the above
We can conclude that we don’t have to check the numbers greater than 4 are the factors of 12.

Think and Grow: Modeling Real Life

Example
There are 4 classes going on a field trip. The classes will use 3 buses. Can the teachers have an equal number of students on each bus?

Think: What do you know? What do you need to find? How will you solve it?
Step 1: Add to find how many students are going on the field trip.
24 + 23 + 25 + 20 = 92
92 students are going on the field trip.
Step 2: Is the total number of students divisible by the number of buses?
Find the sum of the digits of 92. 9 + 2 =11
The sum of the digits not divisible by 3.
The teachers don’t have an equal number of students on each bus.

Show and Grow

Question 17.
A teacher is making a 5-page test with 28 vocabulary problems and 7 reading problems. Can the teacher put an equal number of problems on each page?
Answer: Yes, the teacher can put an equal number of problems on each page

Explanation:
Given that a teacher is making a 5- page test.
It is also given that there are 28 vocabulary problems and 7 reading problems.
So,
Total number of problems = 28 + 7 = 35 problems
So,
The number of problems that each page contains = 35 ÷ 5
Now,
By using the Distributive Property of Multiplication,
35 ÷ 5 = ( 30 + 5 ) ÷ 5
= ( 30 ÷ 5 ) + ( 5 ÷ 5 )
= 6 + 1
= 7
Hence, from the above,
We can conclude that the teacher can put an equal number of problems on each page.

Question 18.
A relay race is 39 laps long. Each team member must bike the same number of laps. Could a team have 8, 6, or 3 members? Explain.
Answer: A team has 3 members.

Explanation:
Given that a relay race is 39 laps long and each member must bike the same number of laps.
Now,
The factors of 39 are: 1, 3, 13, and 39
The factors are the numbers that divide the number originally.
So, 39 can be divided by 3 only.
Hence, from the above,
We can conclude that each team has only 3 members.

Question 19.
DIG DEEPER!
You have 63 clay figures to display on 7 shelves. Not all of the shelves need to be used and each shelf can hold no more than 25 figures. Each shelf must have the same number of figures. What are all the ways you could arrange the figures?

Answer: The number of ways you could arrange the figures is to find the number of factors of 63.
So,
The number of factors of 63 is: 1, 3, 7,9 21, and 63

Explanation:
Given that you have 63 clay figures to display on 7 shelves.
It is also given that each shelf holds no more than 25 figures.
Now,
The factors of  63 are: 1, 3, 7,9 21, and 63
The “Factors” are the ways to arrange the given clay figures.
So,
The number of ways to arrange the clay figures that do not hold more than 25 is: 3, 7, 9, and 21
Hence, from the above,
We can conclude that the number of ways to arrange the clay figures in each shelf that do not hold more than 25 figures is: 3, 7, 9, and 21 ways

Factors and Divisibility Homework & Practice 6.2

Find the factor pairs for the number.
Question 1.
24
Answer:
The factor pairs of 24 are:
1 × 24, 2 × 12, 3 × 8, 4 × 6, 6 × 4, 8 × 3, 12 × 2, and 24 × 1

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factor pairs of 24 are:
1 × 24, 2 × 12, 3 × 8, 4 × 6, 6 × 4, 8 × 3, 12 × 2, and 24 × 1

Question 2.
48
Answer:
The factor pairs of 48 are:
1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factor pairs of 48 are:
1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8

Question 3.
31
Answer:
The factor pairs of 31 are:
1 × 31 and 31 × 1

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factor pairs of 31 are:
1 × 31 and 31 × 1

Question 4.
99
Answer:
The factor pairs of 99 are:
1 × 99, 3 × 33, 9 × 11

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factor pairs of 99 are:
1 × 99, 3 × 33, 9 × 11

Question 5.
45
Answer:
The factor pairs of 45 are:
1 × 45, 3 × 15, and 5 × 9

Explanation:
factors are the numbers that divide the number originally.
Hence,
The factor pairs of 45 are:
1 × 45, 3 × 15, and 5 × 9

Question 6.
26
Answer:
The factor pairs of 26 are:
1 × 26, 2 × 13

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factor pairs of 26 are:
1 × 26, 2 × 13

List the factors of the number.
Question 7.
25
Answer:
The factors of 25 are: 1, 5, and 25

Explanation:
Factors are the numbers that divide the numbers originally.
Hence,
The factors of 25 are: 1, 5, and 25

Question 8.
56
Answer:
The factors of 56 are: 1, 2, 4, 7, 8, 14, 28 and 56

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factors of 56 are: 1, 2, 4, 7, 8, 14, 28 and 56

Question 9.
75
Answer:
The factors of 75 are: 1, 3, 5, 15, 25, and 75

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factors of 75 are: 1, 3, 5, 15, 25, and 75

Question 10.
80
Answer:
The factors of 80 are: 1, 2, 4, 5, 8, 10, 16, 20,40 and 80

Explanation:
Factors are the numbers that divide the numbers originally.
Hence,
The factors of 80 are: 1, 2, 4, 5, 8, 10, 16, 20,40 and 80

Question 11.
93
Answer:
The factors of 93 are: 1, 3, 31 and 93

Explanation;
Factors are the numbers that divide the number originally.
Hence,
The factors of 93 are: 1, 3, 31 and 93

Question 12.
61
Answer:
The factors of 61 are: 1 and 61

Explanation:
Factors are the numbers that divide the number originally.
Hence,
The factors of 61 are: 1 and 61

Question 13.
Reasoning
Why does a number that has 9 as a factor also have 3 as a factor?
Answer: The number that has 9 as a factor also have 3 as a factor because 9 is a multiple of 3

Explanation:
Let the number that has 9 as a factor and that has also 3 as a factor be: 18
Now,
The factors of 18 are: 1, 2, 3, 6, 9, and 18
Hene, from the above,
We can conclude that the number that has 9 as a factor also have 3 as a factor.

Question 14.
DIG DEEPER!
The number below has 3 as a factor. What could the unknown digit be?
3 _____ 5.
Answer: The unknown digit could be: 1 or 4 or 7

Explanation:
Given that the number has 3 as a factor.
To have 3 as a factor, the sum of the digits in the given number should be a multiple of 3
Hence,
The unknown digit in 3____5 could be: 1 or 4 or 7

Question 15.
Number Sense
Which numbers have 5 as a factor?

Answer: The numbers which have 5 as a factor are: 50, 25, 1,485 and 100

Explanation:
Given numbers are: 50, 34, 25, 1,485, 100 and 48
The numbers that have the factor of 5 must have the last digits 0 or 5
Hence,
The numbers that have 5 as a factor are: 50, 25, 1,485, and 100

Question 16.
Modeling Real Life
You and a partner are conducting a bottle flipping experiment. You have 3 bottles with different amounts of water in each. You need to flip each bottle 15 times. If you take turns, will you and your partner each get the same number of flips?
Answer: Yes, you and your partner will each get the same number of flips.

Explanation:
Given that you and your partner are conducting a bottle flipping experiment and you have 3 bottles with different amounts of water in each and you need to flip each bottle 15 times.
So,
The number of flips each will get = 15 ÷ 3
Now,
By using the Distributive Property of Multiplication,
15 ÷ 3 = ( 12 + 3 ) ÷ 3
= ( 12 ÷ 3 ) + ( 3 ÷ 3 )
= 4 + 1
= 5
Hence, from the above,
We can conclude that each bottle will flip 5 times.

Question 17.
Modeling Real Life
A florist has 55 flowers. She wants to put the same number of flowers in each vase without any leftover Should she put 2, 3, or 5 flowers in each vase? Explain.

Answer: The florist put 5 flowers in each vase.

Explanation:
Given that a florist has 55 flowers and she wants to put the same number of flowers in each vase without any leftover.
The number 55 will be divided by to not have any leftover because the last digit is 5.
Hence,
The florist puts 5 flowers in each vase.

Review & Refresh

Compare.
Question 18.

Answer: 7,914 is greater than 7,912

Explanation:
Given numbers are 7,914 and 7,912
Hence, from the above,
We can conclude that 7,914 is greater than 7,912

Question 19.

Answer: 65,901 is less than 67,904

Explanation:
Given numbers are 65,901 and 67,904
Hence, from the above,
We can conclude that 65,901 is less than 67,904

Question 20.

Answer: 839,275 is equal to 839,275

Explanation:
Given numbers are 839,275 and 839,275
Hence, from the above,
We can conclude that 839,275 is equal to 839,275

Lesson 6.3 Relate Factors and Multiples

Explore and Grow

List all factors of 24.
Answer:
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24

Explanation:
Factors are the numbers that divide the numbers originally.
Hence,
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24

List several multiples of each factor. What number appears in each list?
Answer:
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24

Explanation:
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24
Now,
There are no other multiples of 1
The multiples of 2 are: 1, 2
The multiples of 3 are: 1,3
The multiples of 4 are: 1, 2, 4
The multiples of 6 are: 1, 2, 3 and 6
The multiples of 8 are: 1, 2, 4 and 8
The multiples of 12 are: 1, 2, 3, 4, 6, and 12
The multiples of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24
Hence, from the above,
We can conclude that 1 appears in each list

Number Sense
How are factors and multiples related?
Answer:
Factors are the numbers that can divide the numbers originally.
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.

Think and Grow: Identify Multiples

A whole number is a multiple of each of its factors.
12 is a multiple of 1, 2, 3, 4, 6, and 12.
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
4 × 3 = 12
6 × 2 = 12
12 × 1 = 12

Example
Is 56 a multiple of 7?

One Way:
List multiples of 7.
7, 14, 21, 28, 35, 42, 49, 56
So, 56  is a multiple of 7.
Another Way:
Use division to determine whether 7 is a factor of 56.
56 ÷ 7 = 8
7 is a factor of 56.
So, 56 is a multiple of 7.

Example
Is 9 a factor of 64?
One Way:
Use divisibility rules to determine whether 9 is a factor of 64.
9 is not a factor of 64 because 6 + 4 = 10 is not divisible by 9.
Another Way:
List the multiples of 9.
9, 18, 27, 36, 45, 54, 63, 72
64 is not a multiple of 9.
So, 9 is not a factor of 64.

Show and Grow

Question 1.
Is 23 a multiple of 3? Explain.
Answer: 23 is not a multiple of 3

Explanation:

Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27 and 30
Hence, from the above,
We can conclude that 23 is not a multiple of 3.

Question 2.
Is 8 a factor of 56? Explain.
Answer: 8 is a factor of 56.

Explanation:
Factors are the numbers that divide the numbers originally.
Now,
The factors of 56 are: 1, 2, 4, 7, 8, 14, 28, and 56
Hence, from the above,
We can conclude that 8 is a factor of 56.

Apply and Grow: Practice

Question 3.
Is 65 a multiple of 5? Explain.
Answer: 65 is a multiple of 5

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35,40, 45, 50, 55, 60 and 65
Hence, fro the above,
We can conclude that 65 is a multiple of 5.

Question 4.
Is 14 a multiple of 4? Explain.
Answer: 14 is not a multiple of 4.

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, and 40
Hence, from the above,
We can conclude that 14 is not a multiple of 4.

Question 5.
Is 23 a multiple of 2? Explain.
Answer: 23 is not a multiple of 2

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
The multiples of 2 are: 2, 4, 6, 8, 10, 1, 14, 16, 18, 20, 22, 24 and 26
Hence, from the above,
We can conclude that 23 is not a multiple of 2.

Question 6.
Is 6 a factor of 96? Explain.
Answer: 6 is a factor of 96

Explanation:
Factors are the numbers that divide the number originally.
The factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 28, 32, 48 and 96
Hence, from the above,
We can conclude that 6 is a factor of 96.

Question 7.
Is 3 a factor of 82? Explain.
Answer: 82 is not a factor of 3

Explanation:
Factors are the numbers that divide the number originally.
The given number is 82
The sum of digits of 82 = 8 + 2 = 11
The number is a multiple of 3 only then the sum of the digits of that number is a multiple of 3.
But the sum of digits of 82 is not a multiple of 3
Hence, from the above,
we can conclude that 3 is not a factor of 82

Question 8.
Is 9 a factor of 72? Explain.
Answer: 9 is a factor of 72

Explanation:
Factors are the numbers that divide the number originally.
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72
Hence, from the above,
We can conclude that 9 is a factor of 72.

Tell whether 8 is a multiple or a factor of the number. Write multiple, factor or both.
Question 9.
4
Answer: 8 is a multiple of 4.

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
So,
The multiples of 4 are: 4, 8, 12, 16, and 20
Hence, from the above,
We can conclude that 8 is a multiple of 4

Question 10.
8
Answer: 8 is a multiple of 8

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
So,
The multiples of 8 are: 8, 16, 24, 32, 0, etc.
Hence, from the above,
We can conclude that 8 is a multiple of 8

Question 11.
32
Answer: 32 is a multiple of 8

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
So,
The multiples of 8 are: 8, 16, 24, 32, 40, 48 etc
Hence, from the above,
We can conclude that 32 is a multiple of 8

Question 12.
Writing
Use numbers 6 and 12 to explain how factors and multiples are related.
Answer:
The multiplication Expression using numbers 6 and 12 is:
6 × 2 = 12
From the multiplication Expression,
6 is a factor of 12
12 is a multiple of 6

Explanation:
The given numbers re 6 and 12
From the given numbers, the multiplication Expression is:
6 × 2 = 12
Hence, from the above multiplication Expression,
6 is a factor of 12
12 is a multiple of 6.

Question 13.
Complete the Venn diagram.

Answer:

Explanation:
From the given Venn diagram,
The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, and 40
The first 10 multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50
Hence, from the above,
The common numbers from both the factors of 4 and the multiples of 5 are:
5, 10, 20 and 40

Think and Grow: Modeling Real Life

Example
You need 96 balloons for a school dance. Balloons come in packs of 4, packs of 6, and packs of 9. Which packs could you buy so you have no leftover balloons?

Use division to determine whether 96 is a multiple of 4.
4 √96 So, 4  is a factor of 96, and 96  is a multiple of 4.
Use the divisibility rules to check whether 96 is a multiple of 6.
96 is even and 9 + 6 = 15 divisible by 3. So, 6  is a factor of 96 and 96  is a multiple of 6.
Use the divisibility rules to check whether 96 is a multiple of 9.
9 + 6 = 15  is not  divisible by 9. So, 9  is not a factor of 96 and
96  is not a multiple of 9.
You could buy packs of 4 balloons or packs of 6 balloons.

Show and Grow

Question 14.
A teacher needs 88 batteries for science experiments. Batteries are sold in packs of 2, packs of 6, and packs of 8. Which packs could the teacher buy so she has no leftover batteries?

Answer: The teacher should buy the packs of 2 and packs of 8 so that there will be no leftover batteries.

Explanation:
Given that a teacher needs 88 batteries for science experiments and that batteries are sold in packs of 2, 6, and 8.
So,
The factors of 88 are: 1, 2, 4, 8, 11, 22, 44, and 88
So, from the factors of 88, we see that there is no 6 as a factor of 88
So, from this, we can say that we can’t pack 88 batteries in packs of 6 but in the packs of 2 and 8 ( Since 2 and 8 both are the factors of 88 )
We know that,
Factors are the numbers that divide the number originally.
Hence, from the above,
We can conclude that the 88 batteries can be packed in packs of 2 or packs of 8.

Question 15.
DIG DEEPER!
Descartes buys 2 books for a total of $15. Each book costs a multiple of $3. How much could each book cost?

Answer: The cost of each book that Descartes bought is: $15

Explanation:
Given that Descartes bought 3 books for a total of $15 and each book costs a multiple of $3
So,
The cost of each book = The total cost of 2 books ÷ Total number of books
= 15 ÷ 2
Now,
By using the Distributive Property of Multiplication,
15 ÷ 2 = ( 12 + 3 ) ÷ 2
( 12 ÷ 2 ) + ( 3 ÷ 2 )
= 6 + 1.5
= 7.5
Hence, from the above,
We can conclude that the cost of each book is: $7.5 and the cost of each book is also a multiple of $3

Question 16.
Newton buys some boxes of dog treats for $9 each. Descartes buys some bags of cat treats for $6 each. Newton and Descartes spend the same amount of money on treats. What is the least amount of money they could have spent?

Answer: The least amount of money that Newton and Descartes spent on dog treats = $18

Explanation:
Given that Newton buys some boxes of dog treats for $9 each and Descartes buys some bags of cat treats for $6 each.
It is also given that Newton and Descartes spend the same amount of money on treats
So, we have to find the least number that can be a multiple of both 6 and 9.
Now,
the multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, and 60
The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, and 90
From the multiples of 6 and 9,
We can see that the least number that can be the multiple of both 6 and 9 is: 18
Hence, from the above,
We can conclude that the least amount of money that Newton and Descartes spent on dog treats is: $18

Relate Factors and Multiples Homework & Practice 6.3

Question 1.
Is 16 a multiple of 3? Explain.
Answer: 16 is not a multiple of 3

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
So,
The multiples of 16 are 3, 6,9, 12, 15, 18, etc.
Hence, from the above,
We can conclude that 16 is not a multiple of 3.

Question 2.
Is 21 a multiple of 7? Explain.
Answer: 21 is a multiple of 7

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
So,
The multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70
Hence, from the above,
We can conclude that 21 is a multiple of 7

Question 3.
Is 46 a multiple of 2? Explain.
Answer: 46 is a multiple of 2

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
So,
For the multiple of 2, the last digit should be 2, 4, 6, 8, 0
So,
46 can be said as a multiple of 2
Hence, from the above,
We can conclude that 46 is a multiple of 2.

Question 4.
Is 5 a factor of 71? Explain.
Answer: 5 is not a factor of 71

Explanation:
Factors are the numbers that divide the number completely.
So,
The factors of 71 are: 1, 71
Hence, from the above,
We can conclude that 5 is not a factor of 71.

Question 5.
Is 8 a factor of 88? Explain.
Answer: 8 is a factor of 88

Explanation:
Factors are the numbers that divide the numbers completely
So,
The factors of 88 are: 1, 2, 4, 8, 11, 22, 44, 88
Hence, from the above factors,
We can conclude that 8 is a factor of 88

Question 6.
Is 4 a factor of 80? Explain.
Answer: 4 is a factor of 80

Explanation:
Factors are the numbers that divide the number completely.
So,
The factors of 80 are: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
Hence, from the above factors,
We can conclude that 4 is a factor of 80.

Tell whether 30 is a multiple or a factor of the number. Write multiple, factor, or both.
Question 7.
30
Answer: 30 is a multiple and factor of 30

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
Factors are the numbers that divide the number completely
So,
The multiples of 30 are: 30, 60, 90, 120, 150, 180, 210, 240, 270 and 300
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30
Hence,
From the above,
We can conclude that 30 is a multiple and factor of 30

Question 8.
90
Answer: 90 is a multiple of 30

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
So,
The multiples of 30 are: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300
Hence, from the above multiples,
We can conclude that 90 is a multiple of 30

Question 9.
10
Answer: 10 is a factor of 30

Explanation:
Factors are the numbers that divide the number originally.
So,
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30
Hence, from the above factors,
We can conclude that 10 is a factor of 30

Tell whether 10 is a multiple or a factor of the number. Write multiple, factor, or both.
Question 10.
5
Answer: 5 is a factor of 10

Explanation:
Factors are the numbers that divide the number completely.
So,
The factors of 10 are: 1, 2, 5, 10
Hence, from the above factors,
We can conclude that 5 is a factor of 10

Question 11.
60
Answer: 60 is a multiple of 10

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
So,
The multiples of 10 are: 10, 20, 30, 40, 50, 60, 70, 80 , 90, 100
Hence, from the above multiples,
We can conclude that 60 is a multiple of 10

Question 12.
10
Answer: 10 is a multiple and factor of 10

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
Factors are the numbers that divide the number completely
So,
The number of factors of 10 is: 1, 2, 5, 10
The multiples of 10 are: 10, 20, 30, 40, 50, 60, 70, 80 , 90, 100
Hence, from the above,
We can conclude that 10 is a factor and a multiple of 10

Question 13.
DIG DEEPER!
Name two numbers that are each a multiple of both 3 and 4. What do you notice about the two multiples?
Answer: 12 is a multiple of both 3 and 4

Explanation:
We know that,
If the number is divisible by 3, then the sum of the digits of that given number must be divisible by 3
If the number is divisible by 4, then the last 2 digits of that given number must also be divisible by 4
Hence,
The number which is a multiple of both 3 and 4 is: 12
12 will satisfy both the conditions of the number divisible by 3 and 4
Hence, from the above,
We can conclude that 12 is a multiple of both 3 and 4

Question 14.
YOU BE THE TEACHER
Is Newton correct? Explain.

Answer: Yes, Newton is correct

Explanation:
Given that,
According to Newton, all numbers that are multiples of 10 have 2 as a factor.
Now,
The multiples of 10 are: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
So,
Let a multiple of 10 be 20
So,
The factors of 20 are: 1, 2, 4, 5, 10, 20
Hence, from the above,
We can conclude that Newton’s statement is correct

Question 15.
Logic
A quotient is a multiple of 4. The dividend is a multiple of 8. The divisor is a factor of 6. Write one possible equation for the problem.

Answer:
The possible equation for the problem is:
24 ÷ 6 = 4

Explanation:
Given that a quotient is a multiple of 4.
It is also given that the dividend is a multiple of 8 and the divisor is a factor of 6.
So,
The possible Equation for the given problem is:
24 ÷ 6 = 4
Where,
24 is a dividend
6 is a divisor
4 is a quotient
Hence, from the above,
We can conclude that the possible  equation according to the given conditions is:
24 ÷ 6 = 4

Question 16.
Modeling Real Life
Your friend needs our friend needs t0 50 US state capitals. She wants to memorize the same number of capitals each day. Which numbers of capitals can she memorize each day: 2, 3, 4, or 5?

Answer: The number of capitals she can memorize each day in packs of 2 and 5

Explanation:
Given that your friend needs 50 US state capitals and she wants to memorize the same number of capitals each day.
It is also given that the number of capitals she had to memorize in the packs of 2, 3, 4, or 5
So,
If the number has to be divided by 5, the last digit of that given number should be 5 (or) 0
So,
The factors of 50 are: 1, 2, 5,  10, 25, 50
Hence, from the above,
We can conclude that 50 US capitals she had to memorize is in the packs of 2 or packs of 5

Question 17.
Modeling Real Life
Zookeepers plan an enrichment day for the animals every 7 days and bathe the elephants every 2 days. You want to go to the zoo when both events are happening. What other dates in May will this happen?

Answer: The events of both Enrichment day and Elephant Day come in the month of May other than 14 May is:
21 May and 28 May

Explanation:
Given that Zookeepers plan an enrichment day for the animals every 7 days
It is also given that the Elephant bathing takes place every 2 days.
So, other than 14 May,
The other days that these two events will take place on May are: 21 May and 28 May with 7 days gap from 14 May
Hence, from the above,
We can conclude that  both the events are happening on 21 and 28 May

Review & Refresh

Estimate the sum or difference.
Question 18.

Answer: 71,606 – 49,641 = 21,965

Question 19.

Answer: 75,294 + 36,043 = 111,337

Question 20.

Answer: 93,294 – 40,293 = 53,001

Lesson 6.4 Identify Prime and Composite Numbers

Explore and Grow

Draw as many different rectangles as possible that each has the given area. Label their side lengths.

Compare the numbers of factors of 32 and 13.

Answer:
The factors of 32 are: 1, 2, 4, 8, 16, 32
The factors of 13 are: 1, 13
On comparison of the factors of 32 and 13, we can say that 13 has less number of factors than 32

Reasoning
Can a whole number have fewer than two factors? exactly two factors? more than two factors?

Answer: Yes, a whole number has exactly 2 factors and more than 2 factors but not less than 2 factors.

Explanation:
A whole number will be divided into 2 types based on the number of factors. They are:
A) Composite numbers:
The numbers that have more than 2 factors are called “Composite numbers”
B) Prime numbers:
The numbers that have exactly 2 factors are called “Prime numbers”
There will no less than 2 factors for any number.

Think and Grow: Identify Prime and Composite Numbers
A prime number is a whole number greater than 1 with exactly two factors, 1 and itself. A composite number is a whole number greater than 1 with more than two factors.

Example
Tell whether 27 is a prime composite
Use divisibility rules.

• 27 is odd, so it is not divisible by 2 or any other even number.
• 2 + 7 = 9 is divisible by 3,
so 27 is divisible by 3.
27 has factors in addition to 1 and itself.
So, 27 is a Composite number.

Example
Tell whether 11 is a prime composite
Use divisibility rules.
• 11 is odd, so it is not divisible by 2 or any other even number.
• 1 + 1 = 2 is not divisible by 3 or 9,
so 11 is not divisible by 3 or 9.
• The ones digit is not 0 or 5,
so 11 is not divisible by 5.
11 has exactly two factors, 1 and itself.
So, 11 is a prime number.

Show and Grow

Tell whether the number is prime or composite. Explain.
Question 1.
7
Answer: 7 is a prime number

Explanation:
Prime number:
The numbers which have exactly 2 factors 1 and itself is called “Prime numbers”
Now,
The factors of 7 are: 1, 7
Hence, from the above,
We can conclude that 7 is a prime number

Question 2.
12
Answer: 12 is a Composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are called “Composite numbers”
Now,
The factors of 12 are: 1, 2, 3, 4, 6, 12
Hence, from the above,
We can conclude that 12 is a Composite number

Question 3.
2
Answer: 2 is a prime number.

Explanation:
Prime number:
The numbers which have exactly 2 factors 1 and itself are called “Prime numbers”
Now,
The factors of 2 are: 1, 2
Hence, from the above,
We can conclude that 2 is a prime number

Question 4.
19
Answer: 19 is a prime number

Explanation:
Prime number:
The numbers which have exactly 2 factors 1 and itself are called “Prime numbers”
Now,
The factors of 19 are: 1, 19
Hence, from the above,
we can conclude that 19 is a prime number

Question5.
45
Answer: 45 is a Composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are called “Composite numbers”
Now,
The factors of 45 are: 1, 3, 5, 9, 15, 45
Hence, from the above,
we can conclude that 45 is a Composite number

Question 6.
54
Answer 54 is a  Composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are called “Composite numbers”
Now,
The factors of 54 are: 1, 2. 3. 6, 9, 18, 27, 54
Hence, from the above,
We can conclude that 54 is a Composite number

Apply and Grow: Practice

Tell whether the number is prime or composite. Explain.
Question 7.
35
Answer:35 is a Composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are called “Composite numbers”
Now,
The factors of 35 are: 1, 5, 7, 35
Hence, from the above,
We can conclude that 35 is a Composite number

Question 8.
5
Answer: 5 is a prime number

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are called “prime numbers”
Now,
The factors of 5 are: 1, 5
Hence, from the above,
we can conclude that 5 is a prime number

Question 9.
23
Answer: 23 is a prime number

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are called ” Prime numbers”
Now,
The factors of 23 are: 1, 23
Hence, from the above,
We can conclude that 23 is a prime number

Question 10.
40
Answer: 40 is a Composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are called “Composite numbers”
Now,
The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40
Hence, from the above,
We can conclude that 40 is a Composite number

Question 11.
41
Answer: 41 is a prime number

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are called “Prime numbers”
Now,
The factors of 41 are: 1, 41
Hence, from the above,
we can conclude that 41 is a prime number

Question 12.
81
Answer: 81 is a Composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are called “Composite numbers”
Now,
The factors of 81 are: 1, 3, 9, 27, 81
hence, from the above,
we can conclude that 81 is a Composite number

Question 13.
Structure
To create a list of the prime numbers that are less than 100, do the following.

• Place a square around 1. It is neither prime nor composite.
• Circle 2 and cross out all other multiples of 2.
• Circle 3 and cross out all other multiples of 3.
• Circle 5 and cross out all other multiples of 5.
• Circle the next number that is not crossed out. This number is prime. Cross out all other multiples of this number.
• Continue until every number is either circled or crossed out.

What are the prime numbers that are less than 100? Explain why these numbers not were crossed out on the chart.
Answer:
The prime numbers below 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 87, 89, 93, and 97

Think and Grow: Modeling Real Life

Example
A museum volunteer has 76 shark teeth to display. Can the volunteer arrange the teeth into a rectangular array with more than 1 row and more than 1 tooth in each row? Explain.

Use divisibility rules to determine whether 76 is prime or composite.
76 is even, so it is divisible by 2.
76 has factors in addition to 1 and itself.
So, 76 is the Composite number.
So,
The volunteer will arrange the teeth into a rectangular array with more than 1 row and more than 1 tooth in each row.
Show and Grow

Question 14.
A teacher has 29 students in class. Can the teacher separate the students into equal groups? Explain.
Answer: No, the teacher can’t separate the students into equal groups.

Explanation:
Given that a teacher has 29 students in the class
Now,
Let find whether 29 is Prime or Composite
Now,
Factors of 29 are: 1, 29
Hence, from the above factors,
We can conclude that 29 is a prime number
Hence,
The teacher can’t divide 29 students into equal groups.

Question 15.
A band instructor wants to have several ways to organize band members into rectangular arrays on the field for a performance.Should the instructor have 89 members or 99 members on the field? Explain.
Answer: The Instructor should have 99 members on the field so that he can arrange the band members into rectangular arrays.

Explanation:
Given that a band instructor wants to organize band members into rectangular arrays on the field for a performance.
It is also given that the instructor wants to arrange into an array of whether 89 members or 99 members
Now,
Factors of 89 are: 1, 89
Factors of 99 are: 1, 3, 9, 11, 33, 99
Hence, from the above,
Since the 99 members can be arranged into different arrays,
We can conclude that the instructor can arrange the participants into an array of 99 members.

Question 16.
DIG DEEPER!
A paramedic is arranging bandages into 4 bins. An equal number of bandages are in each bin. Did the paramedic arrange a prime number or a composite number of bandages? Explain.

Answer: The paramedic has to arrange a composite number of bandages.

Explanation:
Gien that a paramedic is arranging bandages into 4 bins.
It is also given that there is an equal number of bandages in each bin.
Now,
Given there are 4 bins and 4 is a Composite number
So, 4 can divide only a Composite number but not a prime number.
Prime numbers:
The numbers which have exactly only 2 factors 1 and itself are “Prime numbers”
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Hence, from the above,
We can conclude that the paramedic has to arrange a Composite number of bandages.

Identify Prime and Composite Numbers Homework & Practice 6.4

Tell whether the number is prime or composite. Explain.
Question 1.
3
Answer: 3 is a prime number

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Prime numbers”
Now,
The factors of 3 are: 1, 3
Hence, from the above,
We can conclude that 3 is a prime number

Question 2.
27
Answer: 27 is a composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Now,
The factors of 27 are: 1, 3, 9, 27
Hence, from the above,
We can conclude that 27 is a composite number

Question 3.
46
Answer: 46 is a composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Now,
The factors of 46 are: 1, 2, 23, 46
Hence, from the above,
We can conclude that 46 is a composite number

Question 4.
17
Answer: 17 is a prime number

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Prime numbers”
Now,
The factors of 17 are: 1, 17
Hence, from the above,
We can conclude that 17 is a prime number

Question 5.
53
Answer: 53 is a prime number

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Composite numbers”
Now,
The factors of 53 are: 1, 53
Hence, from the above,
we can conclude that 53 is a prime number

Question 6.
63
Answer: 63 is a composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Now,
The factors of 63 are: 1, 3, 7, 9, 21, 63
Hence, from the above,
We can conclude that 63 is a composite number

Question 7.
29
Answer: 29 is a prime number

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Prime numbers”
Now,
The factors of 29 are: 1, 29
Hence, from the above,
We can conclude that 29 is a prime number

Question 8.
31
Answer: 31 is a prime number

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Prime numbers”
Now,
The factors of 31 are: 1, 31
Hence, from the above,
We can conclude that 31 is a prime number

Question 9.
75
Answer: 75 is a composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Now,
The factors of 75 are: 1, 3, 5, 15, 25, 75
Hence, from the above,
We can conclude that 75 is a composite number

Question 10.
DIG DEEPER!
Can a number be both prime and composite? Explain.
Answer: No, a number can’t be both prime and composite

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Prime numbers”
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Hence, from the number of factors,
We can say a number can’t be both prime and composite at the same time

Question 11.
Logic
Your friend is thinking of a prime number between 60 and 80. The tens digit is one less than the ones digit. What is the number?
Answer: The number is 67

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Prime numbers”
So,
The prime numbers between 60 and 80 are: 61, 67, 73, 79
It is also given that the tens digit is one less than the ones digit
So,
Tens position value – 1 = ones position value
Hence, from the above,
We can conclude that 67 is the number.

Number Sense
Write true or false for the statement. If false, provide an example to support your answer.
Question 12.
All odd numbers are prime. _______
Answer: False

Explanation:
Given that all odd numbers are prime.
Prime numbers:
The number of factors which have exactly 2 factors 1 and itself is “Prime numbers”
So, let take the off numbers from 1 to 10
The odd numbers from 1 to 10 are: 1, 3, 5, 7, 9
So, from 1 to 10,
The prime numbers are: 3, 5, 7
So, from the above
9 is an odd number but it is not prime.
Hence, from the above,
We can conclude that all odd numbers are not prime

Question 13.
All even numbers, except 2, are composite. _______
Answer: True

Explanation:
Given that all even numbers except 2 are composite numbers
Now,
The even numbers from 1 to 0 are: 2, 4, 6, 8
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
So,
The composite numbers from 1 to 10 are: 4, 6, 8
Note: 2 is an even prime number
Hence, from the above,
WE can conclude that all even numbers except  are composite numbers
Question 14.
A composite number cannot have exactly three factors. _______
Answer: True

Explanation:
Given that a composite number has exactly 3 factors
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Let the number be 63
Now,
The factors of 63 are: 1, 3, 7, 9, 21, 63
Hence, from the above,
We can conclude that a composite number don’t have exactly 3 factors

Question 15.
Modeling Real Life
There are 43 students trying out for a basketball team. Can the coach separate the students into equal groups? Explain.

Answer: No, the coach can’t separate the students into equal groups

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Prime numbers”
Now,
The factors of 43 are: 1, 43
Hence, from the above,
We can conclude that the coach can’t separate the students into equal groups since 43 is a prime number

Question 16.
Modeling Real Life
Which planet not does have a prime number of rings?

Answer: Saturn does not have a prime number of rings.

Explanation:
Given that,
1 full circle = 2 rings
So,
1 half circle = 1 ring

So, from the above table,
The number of rings of Jupiter is: 3
The number of rings of Saturn is: 9
The number of rings of Uranus is: 13
The number of rings of Neptune is: 5
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Prime numbers”
Hence, from the above,
We can conclude that Saturn does not have a prime number of rings ( Because 9 is a composite number )

Review & Refresh

Use properties to find the product. Explain your reasoning
Question 17.
4 × 9 × 25
Answer: 4 × 9 × 25 = 900

Explanation:
Using the Distributive Property of multiplication,
4 × 9 × 25 = 4 × 9 × ( 20 + 5 )
= 36 × ( 20 + 5 )
= ( 36 × 20 ) + ( 36 × 5 )
= 720 + 180
= 900
Hence, 4 × 9 × 25 = 900

Question 18.
405 × 3
Answer: 403 × 3 = 1,215

Explanation:
Using the Distributive Property of Multiplication,
405 × 3 = ( 400 + 5 ) × 3
= ( 400 × 3 ) + ( 5 × 3 )
= 1,200 + 15
= 1,215
Hence, 405 × 3 = 1,215

Question 19.
698 × 7
Answer: 698 × 7 = 4,886

Explanation:
Using the Distributive Property of Multiplication,
698 × 7 = ( 600 + 90 + 8 ) × 7
= ( 600 × 7 ) + ( 90 × 7 ) + ( 8 × 7 )
= 4,200 + 630 + 56
= 4,886
Hence, 698 × 7 = 4,886

Lesson 6.5 Number Patterns

Explore and Grow

Shade every third square in the table.

Write the shaded numbers. What patterns do you see?

What other patterns do you see in the table?

Answer:
The shaded numbers are:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60
From the shaded numbers, we can see that each third square is a multiple of 3
The other patterns we can observe in the given table is:
a) If we first shaded the second square and then shaded each second square, then we get the multiples of 2
b) The same pattern will have to be applied for the fourth square
There will also be other patterns by shading 5th square, 6th square, etc.

Structure
Circle every fourth square in the table. Write the circled numbers. What patterns do you see?
Answer:
The circled numbers are:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60
From the circled numbers of every fourth square, we can see that every fourth square is a multiple of 4.

Think and Grow: Create Number Patterns

A rule tells how numbers or shapes in a pattern are related.
Example
Use the rule “Add 3.” to create a number pattern. The first number in the pattern is 3. Then describe another feature of the pattern.
Create a pattern.

So,
The numbers in the pattern are multiples of 3.

Example
Use the rule “Multiply by 2.” to create a number pattern. The first number in the pattern is 10. Then describe another feature of the pattern.
Create a pattern.

So,
The one’s digit of each number in the pattern is 0.

Show and Grow

Write the first six numbers in the pattern. Then describe another feature of the pattern.
Question 1.
Rule: Add 5.
First number: 1
1, ____, _____, _____, _____, _____
Answer: The first 6 numbers are: 1, 6, 11, 16, 21 and 26

Explanation:
For the formation of the pattern,
The given rules are:
Rule 1 : Add 5
Rule 2: Add 5
Hence,
The given pattern is

Question 2.
Rule: Multiply by 3.
First number: 3
3, _____, _____, ____, _____, _____
Answer: The first 6 numbers are: 3, 9, 27, 81, 243, 729

Explanation:
For the formation of the pattern,
the given rules are:
Rule 1: Multiply by 3
Rule 2: First number: 3
Hence,
The given pattern is

Question 3.
Rule: Subtract 2.
First number: 20
Answer: The first 6 numbers are: 20, 18, 16, 14, 12, 10

Explanation:
For the formation of the pattern,
the given rules are:
Rule 1: Subtract 20
Rule 2: First number: 20
Hence,
The given pattern is:

Question 4.
Rule: Divide by 2.
First number: 256
Answer: The first 6 numbers are: 256, 128, 64, 32, 16, 8

Explanation:
For the formation of the pattern,
the given rules are:
Rule 1: Divide by 2
Rule 2: First number: 256
Hence,
The given pattern is:

Apply and Grow: Practice

Write the first six numbers in the pattern. Then describe another feature of the pattern.
Question 5.
Rule: Add 11.
First number: 11
Answer: The first 6 numbers are: 11, 22, 33, 44, 55, 66

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Add 11
First number: 11
Hence,
The given pattern is:

Question 6.
Rule: Multiply by 4.
First number: 4
Answer: The first 6 numbers are: 4, 16, 64, 256, 1024, 4096

Explanation:
For the formation of the pattern,
the given rules are:
Rule 1: Multiply by 4
First number: 4
Hence,
The given pattern is

Question 7.
Rule: Subtract 3.
First number: 21
Answer: The first 6 numbers are: 21, 18, 15, 12, 9, 6

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Subtract 3
First number: 21
Hence,
The given pattern is:

Question 8.
Rule: Divide by 3.
First number: 729
Answer: The first 6 numbers are: 729, 243, 81, 27, 9, 3

Explanation:
For the formation of the pattern,
the given rules are:
Rule 1: Divide by 3
First number: 729
Hence,
The given pattern is

Question 9.
Rule: Add 9.
First number: 8
Answer: The first  numbers are: 8, 17, 26, 35, 44, 53

Explanation:
For the formation of the pattern,
the given rules are:
Rule 1: Add 9
First number: 8
Hence,
The given pattern is:

Question 10.
Rule: Multiply by 5.
First number: 5
Answer: The first 6 numbers are: 5, 25, 125, 625, 3125, 15,625

Explanation:
For the formation of the pattern,
the given rules are:
Rule 1: Multiply by 5
First number: 5
Hence,
The given pattern is:

Open-Ended
Use the rule to generate a pattern of four numbers.
Question 11.
Rule: Multiply by 2.
Answer:
Let the first number be 2.
Hence,
The pattern of four numbers is: 2, 4, 8, 16

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Multiply by 2
Let the first number be: 2
Hence,
The obtained pattern will be:

Question 12.
Rule: Subtract 9.
Answer:
Let the first number be 36
Hence,
The pattern of four numbers is: 36, 27, 18, 9

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Subtract 9
Let the first number be 36
Hence,
The obtained pattern will be:

Question 13.
Rule: Divide by 4.
Answer:
Let the first number be 16
Hence,
The pattern of 4 numbers are: 16, 12, 8, 4

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Divide by 4
Let the first number be 16
Hence,
The obtained pattern will be:

Question 14.
Rule: Add 7.
Answer:
Let the first number be 7
Hence,
The pattern of the four numbers are: 7, 14, 21, 28

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Add 7
Let the first number be 7
Hence,
The obtained pattern will be:

Question 15.
Patterns
Write a rule for the pattern below. Then write a different pattern that follows the same rule.
3, 6, 12, 24, 48
Answer:  The rule for the given pattern is: Multiply by 2

Explanation:
Given numbers are: 3, 6, 12, 24, 48
So, from the given numbers,
We can say that the given pattern follows the “Multiply by 2” rule

Question 16.
Reasoning
What is the missing number in the pattern? Explain.
39, 37, 35, _____, 31, 29
Answer: The missing number in the pattern is: 33

Explanation:
The given numbers are: 39, 37, 35, 31, 29
From the given pattern,
the numbers that are following the rule is “Subtract by 2”
Hence, from the above,
The missing number will be: 35 – 2 = 33

Think and Grow: Modeling Real Life

Example
A presidential election is held every 4 years. There was a presidential election in 2016. How many presidential elections will occur between 2017 and 2030?

The rule is to add 4 years to each presidential election year. Start with 2016. Then count the years in the pattern that is between 2017 and 2030.

So,
3 presidential elections will occur between 2017 and 2030.

Show and Grow

Question 17.
The pattern of animals on a Chinese calendar repeats every 12 years. The year 2000 was the year of the dragon. How many times will the year of the dragon occur between 2001 and 2100?

Answer: The number of times the dragon will appear between 2001 and 2100 is: 8

Explanation:
Given that the pattern of animals on a Chinese calendar repeats every 12 years
It is also given that the year 2000 was the year of the dragon
Now, between the year 2001 and 2100, there are 99 years
So,
The number of times the dragon will appear = The difference between the years 2001 and 2100 ÷ The number of years the pattern will change
= 99 ÷ 12
Now,
99 ÷ 12 = 8 R 3
Hence, from the above,
We can conclude that the dragon will appear 8 times between the years 2001 and 2021

Question 18.
A robotics team raised $25 the first month of school. Each month of school, the team wants to raise 2 times as much money as the month before. How much money should they raise in the fifth month of school?
Answer: The money should they raise in the fifth month of school is: $400

Explanation:
Given that a robotics team raised $25 for the first month of the school and each month of the school, the team wants to raise 2 times as much money as the month before.
Hence,
The rule followed here is: Multiply by 2
The given first number is: $25
Hence,
The pattern we will obtain is:

Hence, from the above,
We can conclude that the money they should raise in the fifth month is: $400

Question 19.
DIG DEEPER!
You start with 128 pictures on your tablet. You take 6 pictures and delete 3 pictures each day. How many pictures do you have on your tablet after 6 days?
Answer: The pictures you have on your tablet after 6 days is: 110 pictures

Explanation:
Given that you have 128 pictures on your tablet and you take 6 pictures and delete 3 pictures each day.
So,
The number of pictures you have each day = 6 – 3 = 3 pictures
So,
The total number of pictures in 6 days = 6 × 3 = 18 pictures
So,
The total number of pictures you have in 6 days = Total number of pictures – Total number f pictures in 6 days
= 128 – 18
= 110 pictures
Hence, from the above,
We can conclude that there are 110 pictures you have in 6 days.

Number Patterns Numbers Homework & Practice 6.5

Write the first six numbers in the pattern. Then describe another feature of the pattern.
Question 1.
Rule: Subtract 8.
First number: 88
88, ____, ____, ____, _____, _____
Answer:
The first 6 numbers in the given pattern are: 88, 80, 72, 64, 56, 48

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Subtract 8
First number: 88
Hence,
The pattern we will obtain is:

Question 2.
Rule: Multiply by 10.
First number: 2
2, _____, _____, _____, ____, ____
Answer:
The first 6 numbers of the given pattern are: 2, 20, 200, 2,000, 20,000, 200,000

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Multiply by 10
First number: 2
Hence,
The pattern we will obtain is:

Question 3.
Rule: Add 9.
First number: 17
Answer:
The first 6 numbers for the given pattern is: 17, 26, 35, 44, 53, 62

Explanation:
For the formation,
the given rules are:
Rule: Add 9
First number: 17
Hence,
The pattern we will obtain is:

Question 4.
Rule: Divide by 2.
First number: 1,600
Answer:
The first 6 numbers of the given pattern are: 1,600, 800, 400, 200, 100, 50

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Divide by 2
First number: 1,600
Hence,
The pattern we will obtain is:

Open-Ended Use the rule to generate a pattern of four numbers.
Question 5.
Rule: Divide by 5.
Answer:
Let the first number be: 625
Hence,
The  four numbers of the given pattern will be: 625, 125, 25, 5

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Divide by 5
Let the first number be: 625
Hence,
The pattern we will obtain is:

Question 6.
Rule: Add 8.
Answer:
Let the first number be 8
Hence,
The four numbers of the given pattern are: 8, 16, 24, 32

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Add 8
Let the first number be 8
Hence,
The pattern we will obtain will be:

Question 7.
Rule: Multiply by 9.
Answer:
Let the first number be 9
Hence,
The four numbers of the pattern will be: 9, 81, 729, 6551

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Multiply 9
Let the first number be 9
Hence,
The pattern we obtain will be:

Question 8.
Rule: Subtract 3.
Answer:
Let the first number be 27
Hence,
The four numbers of the given pattern will be: 27, 24, 21, 18

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Subtract 3
Let the first number be 27
Hence,
The pattern we obtain will be:

Question 9.
Structure
List the first ten multiples of 9. What patterns do you notice with the digits in the one’s place? in the tens place?
Does this pattern continue beyond the tenth number in the pattern?
Answer: The first 10 multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

Explanation:
The first 10 multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
So,
from the above 10 multiples of 9,
The pattern we can observe in the one’s place is: Decreasing of numbers from 9 to 0 i.e., 9, 8, 7, ………..0
The pattern we can observe in the ten’s place is: Increasing of numbers from 0 to 9 i.e., 0, 1, 2 …………..9
The above 2 patterns will continue beyond the first 10 multiples of 9
Hence, from the above,
We can conclude that the patterns we observed are:
The pattern we can observe in the one’s place is: Decreasing of numbers from 9 to 0 i.e., 9, 8, 7, ………..0
The pattern we can observe in the ten’s place is: Increasing of numbers from 0 to 9 i.e., 0, 1, 2 …………..9

Question 10.
Modeling Real Life
It takes the moon about 28 days to orbit Earth. How many times will the moon orbit Earth in 1 year?

Answer: The moon will orbit the earth 10,220 times in 1 year

Explanation:
Given that it takes the moon about 28 days to orbit the earth.
We know that,
1 year = 365 days
So,
The number of times the moon will orbit around the earth in 1 year = 365 × 28
Now,
By using the partial products method,
365 × 28  = ( 300 + 65 ) ×  ( 20 + 8 )
= ( 300 ×  20 ) + ( 300 ×  8 ) + ( 65 ×  20 ) + ( 65 ×  8 )
= 6,000 + 2,400 + 1,300 + 520
= 10,220 times
Hence, from the above,
We can conclude that the moon will orbit around the Earth 10,220 times

Question 11.
DIG DEEPER!
In each level of a video game, you can earn up to 10 points and lose up to 3 points. Your friend earns 9 points in the first level. If he earns and loses the maximum number of points at each level, how many total points will he have after level 6?
Answer: The total points your friend will have is: 44 points

Explanation:
Given that in each level of a video game, you can earn up to 10 points and lose up to 3 points
So,
The maximum number of points you can get = 10 – 3 = 7 points
It is also given that your friend earns 9 points in the first level
So,
The number of points he will have at level 6 = The number of points he has at the first level + 5 × ( The maximum number of points he can get )
= 9 + ( 5 ×  7 )
= 9 + 35
= 44 points
Hence, from the above,
We can conclude that he will have 44 points at the sixth level.

Review & Refresh

Find the product.
Question 12.
14 × 23 = _____
Answer: 14 ×  23 = 322

Explanation:
By using the partial products method,
14 ×  23 = ( 10 + 4 ) × ( 20 + 3 )
= ( 10 ×  20 ) + ( 10 ×  3 ) + ( 4 ×  20 ) + ( 4 ×  3 )
= 200 + 30 + 80 + 12
= 322
Hnece, 14 × 23 = 322

Question 13.
48 × 60 = _____
Answer: 48 × 60 = 2,880

Explanation:
By using the Distributive Property of Multiplication,
48 ×  60 = ( 40 + 8 ) × 60
= ( 40 ×  60 ) + ( 8 ×  60 )
= 2,400 + 480
= 2,880
Hence, 48 × 60 = 2,880

Question 14.
55 × 31 = _____
Answer: 55 ×  31 = 1,705

Explanation:
By using the partial products method,
55 × 31 = ( 50 + 5 ) ×  ( 30 + 1 )
= ( 50 ×  30 ) + ( 50 ×  1 ) + ( 5 ×  30 ) + ( 5 ×  1 )
= 1,500 + 50 + 150 + 5
= 1,705
Hence, 55 ×  31 = 1,705

Lesson 6.6 Shape Patterns

Explore and Grow

Create a rule using 3 different shapes. Draw the first six shapes in the pattern.

What is the next shape in the pattern?

What is the 9th shape in the pattern? Explain.

What is the 99th shape? 1,000th shape? Explain?

Answer: 
From the above 3 shapes,
The rule we can create is: Add 1
So,
We can say the above pattern will repeat after every 3 shapes.
So,
The 9th shape will be: Pentagon
The 99th shape will be: 99 / 3  =  33 R 0
So, The 99th shape will be: Pentagon
The 100th shape will be: Triangle (Because the remainder will be 1 i.e., the shapes will completely repeat 99 times and for the 100th time, it will start from 1st shape)

Structure
You want to show the first 40 shapes in the pattern above. Without modeling, how many of each shape do you think you will need?
Answer:
The number of times each shape will appear in the first 40 shapes is:
Triangle: 14 times
Square: 13 times
Pentagon: 13 times

Explanation:
In the above pattern, the shapes are: Triangle, Square, Pentagon
The rule we obtained by using these 3 shapes is: Add 1
So,
The pattern for the first 40 shapes will be: 40 ÷ 3
Now,
By using the Distributive Property of Division,
40 ÷ 3 = ( 36 + 3 ) ÷ 3
= ( 36 ÷ 3 ) + ( 3 ÷ 3 )
= 12 + 1
= 13 R 1
Hence, the 3 shapes will appear 39 times
So,
The number of times each shape will appear = 13 times
Hence, from the above,
We can conclude that the number of times the shape will appear is:
Triangle: 14 times ( 13 + 1. This 1 is because 40th time, the triangle will repeat again )
Square: 13 times
Pentagon: 13 times

Think and Grow: Create Shape Patterns

Example
Create a shape pattern by repeating the rule “triangle, hexagon, square, rhombus.”What is the 42nd shape in the pattern?
Create a pattern.

42 ÷ 4 is 10 R2, so when the pattern repeats 10 times, the 40th shape is a rhombus So, the 41st shape is a Triangle and the 42nd shape is a hexagon.


Figure 1 has 1 column of 4 dots, so it has 1 × 4 = 4 dots.
Figure 2 has 2 columns of 4 dots, so it has 2 × 4 = 8 dots.
Figure 3 has 3 columns of 4 dots, so it has 3 × 4 = 12 dots.
So,
The 25th figure has 1 column of 4 dots, ( Because the 3 figures will be repeated 24 times and the first figure will appear again 25th time)
So, it has 1 × 4 =4 dots.

Show and Grow

Question 1.
Extend the pattern of shapes by repeating the rule “square, trapezoid, triangle, hexagon, triangle.”What is the 108th shape in the pattern?

Answer: The 108th shape in the pattern will be: Triangle

Explanation:
The given pattern is: Square, trapezoid, triangle, hexagon, triangle
So, there is a total of 5 shapes.
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
108 ÷ 5 = ( 100 + 5 ) ÷ 5
= ( 100 ÷ 5 ) + ( 5 ÷ 5 )
= 20 + 1
= 21 R 3
Hence, from the above,
We can conclude that the 108th shape will be the triangle.

Question 2.
Describe the dot pattern. How many dots are in the 76th figure?

Answer: The number of dots in the 76th figure is: 3 dots

Explanation:

From the above pattern,
In figure 1, the number of dots = 1 × 3 = 3 dots
In figure 2, the number of dots = 2 × 3 =  dots
In figure 3, the number of dots = 3 × 3 = 9 dots
So,
The total number of patterns is: 3
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
76 ÷ 3 = ( 66 +9 ) ÷ 3
= ( 66 ÷ 3 ) + ( 9 ÷ 3 )
= 22 + 3
= 25 R 1
Hence, from the above,
We can conclude that there are 3 dots in the 76th figure.

Apply and Grow: Practice
Question 3.
Extend the pattern of shapes by repeating the rule “oval, triangle.”What is the 55th shape?

Answer: The 55th shape is: Oval

Explanation:
The given pattern of shapes is: Oval, Triangle
So,
The total number of patterns is: 2
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
55 ÷ 2 = ( 50 + 4 ) ÷ 2
= ( 50 ÷ 2 ) + ( 4 ÷ 2 )
= 25 + 2
= 27 R 1
Hence, from the above,
We can conclude that the 55th shape is: Oval

Question 4.
Extend the pattern of symbols by repeating the rule “add, subtract, multiply, divide.”What is the 103rd symbol?

Answer: The 103rd symbol is: Multiply

Explanation:
The given patterns are: Add, Subtract, Multiply, Divide
So,
The total number of patterns is: 4
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
103 ÷ 4 = ( 80 + 20 ) ÷ 4
= ( 80 ÷ 4 ) + ( 20 ÷ 4 )
= 20 + 5
= 25 R 3
Hence, from the above,
We can conclude that the 103rd shape is: Multiply

Question 5.
Describe the pattern. How many squares are in the 24th figure?

Answer: The number of squares in the 24th figure is: 4 squares

Explanation:

In figure 1, the number of squares is: 2
In figure 2, the number of squares is: 3
In figure 3, the number of squares is: 4
So,
Total number of figures = 3
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
24 ÷ 3 = ( 21 + 3 ) ÷ 3
= ( 21 ÷ 3 ) + ( 3 ÷ 3 )
= 7 + 1
= 8 R 0
Hence, from the above,
We can conclude that  in the 24th figure, the number of squares is: 4

Question 6.
Describe the pattern of the small triangles. How many small triangles are in the 10th figure?

Answer: The small triangles in the 10th figure is: 1

Explanation:

From the given figure,
In fig 1, the number of triangles is: 1
In fig 2, the number of triangles is: 2
In fig 3, the number of triangles is: 4
So,
Total number of figures = 3
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
10 ÷ 3 = ( 3 + 6 ) ÷ 9
= ( 3 ÷ 3 ) + ( 6 ÷ 3 )
= 1 + 2
= 3 R 1
Hence, from the above,
We can conclude that the 10th figure has 1 triangle

Question 7.
Structure
Make a shape pattern that uses twice as many squares as triangles.
Answer:

Question 8.
Number Sense
Which shape patterns have a heart as the 12th shape?

Answer:
Let the patterns be named as A), B), C) and D)
So,
The shape patterns that have a heart as the 12th shape is: A), B) and D)

Explanation:

Let the patterns be named as A), B), C) and D)
A) In pattern A,
The total number of figures = 2
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
12 ÷ 2 = 6
Hence, A) will have a heart as the 12th shape
B) In pattern B,
The total number of figures = 4
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
12 ÷ 4 = 3
Hence, B) will have a heart as the 12th shape
C) In pattern C,
The total number of figures = 2
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
12 ÷ 2 = 6
But, the 12th shape in C) will be a circle
D) In pattern D,
The total number of figures = 3
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
12 ÷ 3 = 4
Hence, C) will have a heart as the 12th shape

Think and Grow: Modeling Real Life

Example
You make a necklace with a cube, hexagon, and star beads. You string the beads in a pattern. You use the rule “cube, star, cube, hexagon.”It takes 64 beads to complete the necklace. How many times do you repeat the pattern?

Divide the number of beads it takes to complete the necklace by the number of beads in the rule. There are 4 beads in the rule.
So,
4√64 = 64 ÷ 4
By using the Distributive Property of division,
64 ÷ 4 = ( 60 + 4 ) ÷ 4
= ( 60 ÷ 4 ) + ( 4 ÷ 4 )
= 15 + 1
= 16
Hence,
You repeat the pattern 16 times

Show and Grow

Question 9.
The path on a board game uses the rule “red, green, pink, yellow, blue.”There are 55 spaces on the game board. How many times does the pattern repeat?

Answer: The number of times the pattern will repeat is: 11 times

Explanation:
Given that the path on a board game uses the rule ” red, green, pink, yellow, blue ”
It is also given that there are 55 spaces on the game board
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the Distributive Property of division,
55 ÷ 5 = ( 50 + 5 ) ÷ 5
= ( 50 ÷ 5 ) + ( 5 ÷ 5 )
= 10 + 1
= 11
Hence, from the above,
We can conclude that the number of times the patterns repeat is: 11 times

Question 10.
You make a walkway in a garden using different-shaped stepping stones. You use the rule “square, circle, square, hexagon.”You use 24 square stepping stones. How many circle and hexagon stepping stones do you use altogether? How many stones do you use in all?
Answer:
The Total number of stones used are: 24
The number of  circle and hexagon stepping stones use are: 12

Explanation:
Given that there is a walkway in a garden using different-shaped stepping stones.
The rule used here is: Square, circle, square, hexagon
It is also given that there are 24 stepping stones you used
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the Distributive Property of division,
24 ÷ 4 = ( 20 + 4 ) ÷ 4
= ( 20 ÷ 4 ) + ( 4 ÷ 4 )
= 5 + 1
= 6
So,
The number of stones used each time is: 6 times
So,
The number of  circle and hexagon stepping stones used altogether is: 6 + 6 = 12
Hence, from the above,
We can conclude that circle and hexagon stepping stones you used altogether is: 12

Question 11.
DIG DEEPER!
You make a rectangular picture frame using square tiles. The picture frame is 12 tiles long and 8 tiles wide. You arrange the tiles in a pattern. You use the rule “red, orange, yellow.” How many of each color tile do you use?
Answer: The number of each color tiles do you use is: 32

Explanation:
Given that you make a rectangular picture frame using square tiles and it is also given that the picture frame is 12 tiles long and 8 tiles wide.
So,
The area of a rectangular picture frame = 12 × 8 = 96 square- meters
The rule used is: red, orange, yellow
So,
The total number of colors is: 3
So,
The number of each color tile you used = Area of rectangular picture frame ÷ Total number of colors
= 96 ÷ 3
Now,
By using the Distributive Property of division,
96 ÷ 3 = ( 90 + 6 ) ÷ 3
= ( 90 ÷ 3 ) + ( 6 ÷ 3 )
= 30 + 2
= 32
Hence, from the above,
we can conclude that each number of color tile used is: 32

Shape Patterns Numbers Homework & Practice 6.6

Question 1.
Extend the pattern of shapes by repeating the rule “up, right, down, left.”What is the 48th shape in the pattern?

Answer: The 48th shape in the pattern is: Left

Explanation:
Given pattern is:

The rule for the given pattern is: up, right, down, left
So,
The total number of patterns is: 4
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
48 ÷ 4 = ( 40 + 8 ) ÷ 4
= ( 40 ÷ 4 ) + ( 8 ÷ 4 )
= 10 + 2
= 12 R 0
Hence, from the above
We can conclude that the 48th figure in the given pattern is: Left

Question 2.
Extend the pattern of shapes by repeating the rule “small circle, medium circle, large circle.”What is the 86th shape in the pattern?

Answer: The 86th shape in the pattern is: Medium circle

Explanation:
Given pattern is:

The rule for the pattern is: Small circle, Medium circle, large circle
So,
The total number of patterns = 3
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
86 ÷ 3 = ( 75 + 9 ) ÷ 3
= ( 75 ÷ 3 ) + ( 9 ÷ 3 )
= 25 + 3
= 28 R 2
Hence, from the above,
We can conclude that the 86th shape in the pattern is: Medium circle

Question 3.
Describe the dot pattern. How many dots are in the 113th figure?

Answer: The number of dots in the 113th figure is: 4 dots

Explanation:
Given pattern is:

So,
In fig 1, the number of dots = 2
In fig 2, the number of dots = 4
In fig 3, the number of dots = 6
So,
The total number of figures = 3
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
113 ÷ 3 = ( 99 + 12 ) ÷ 3
= ( 99 ÷ 3 ) + ( 12 ÷ 3 )
= 33 + 4
= 37 R 2
Hence, from the above,
we can conclude that the number of dots in the 113th figure is: 4 dots

Question 4.
YOU BE THE TEACHER
You and your friend each create a shape pattern with 100 shapes. Your friend says both patterns will have the same number of circles. Is your friend correct? Explain.

Answer: Yes, both patterns will have the same number of circles.

Explanation:
Given patterns are:

The pattern of the friend is: Star, circle
The pattern of you si: Heart, circle, pentagon, circle
It is also given that,
The total number of figures = 100
So,
The total number of figures for your friend = 2
The total number of figures for you = 4
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So
For your friend,
By using the partial quotients method,
100 ÷ 2 = ( 50 + 50 ) ÷ 2
= ( 50 ÷ 2 ) + ( 50 ÷ 2 )
= 25 + 25
= 50
Hence,
The number of circles = 50
The number of stars = 50
For you,
By using the partial quotients method,
100 ÷ 4 = ( 80 + 20 ) ÷ 4
= ( 80 ÷ 4 ) + ( 20 ÷ 4 )
= 20 + 5
= 25
Hence,
The number of circles altogether = 25 + 25 = 50
Hence, from the above,
We can conclude that there are equal number of circles in both the patterns

Question 5.
Structure
Draw the missing figure in the pattern. Explain the pattern.

Answer:

Question 6.
Reasoning
Newton uses the rule “bone, bone, paw print” to make a shape pattern. He wants the pattern to repeat 8 times. How many bones will be in Newton’s pattern?

Answer: The number of bones in Newton’s pattern is: 16

Explanation:
Given pattern is:

The rule followed in the pattern is: Bone, Bone, pawprint
So,
The total number of figures in the given pattern = 3
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
It is also given that the pattern repeated 8 times.
So,
The total number of figures when the pattern repeated 8 times = 8 × 3 = 24 figures
So,
By using the partial quotients method,
24 ÷ 3 = ( 21 + 3 ) ÷ 3
= ( 21 ÷ 3 ) + ( 3 ÷ 3 )
= 7 + 1
= 8
So,
The number of each figure in a pattern = 8
From the pattern, we can say there are 2 bones
So,
The total number of bones when the pattern repeated 8 times = 2 × 8 = 16
Hence, from the above,
We can conclude that there are 16 bones in the pattern when the pattern repeated 8 times

Question 7.
Modeling Real Life
The black keys on a piano follow the pattern “two black keys, three black keys.” There are 36 black keys on a standard piano. How many times does this entire pattern repeat?
Answer: The entire pattern repeat 12 times

Explanation:
Given that the black keys on a pattern follow the pattern
The given pattern is: two black keys, three black keys
So,
The total number of patterns = 2
It is also given that there are 36 black keys on a standard piano.
So,
The total number of black keys on a piano = 36
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
36 ÷ 2 = ( 30 + 6 ) ÷ 2
= ( 30 ÷ 2 ) + ( 6 ÷ 2 )
= 15 + 3
= 18
Hence, from the above,
We can conclude that the entire pattern will repeat 18 times.

Review & Refresh

Find the quotient.
Question 8.
30 ÷ 5 = _____
Answer: 30 ÷ 5 = 6

Explanation:
By using the partial products method,
30 ÷ 5 = ( 25 + 5 ) ÷ 5
= ( 25 ÷ 5 ) + ( 5 ÷ 5 )
= 5 + 1
= 6
Hence, 30 ÷ 5 = 6

Question 9.
360 ÷ 9 = ______
Answer: 360 ÷ 9 = 40

Explanation:
By using the partial quotients method,
360 ÷ 9 = ( 270 + 90 ) ÷ 9
= ( 270 ÷ 9 ) + ( 90 ÷ 9 )
= 30 + 10
= 40
Hence, 360 ÷ 9 = 40

Question 10.
6,400 ÷ 8 = _____
Answer: 6,400 ÷ 8 = 800

Explanation:
By using the place -value method,
6,400 ÷ 8 = 64 hundreds ÷ 8
= 8 hundred
= 8 × 100
= 800
Hence, 6,400 ÷ 8 = 800

Question 11.
140 ÷ 2 = _____
Answer: 140 ÷ 2 = 70

Explanation:
By using the place-value method,
140 ÷ 2 = 14 tens ÷ 2
= 7 tens
= 7 × 10
= 70
Hence, 140 ÷ 2 = 70

Question 12.
4,200 ÷ 7 = _____
Answer: 4,200 ÷ 7 = 600

Explanation:
By using the place-value method,
4,200 ÷ 7 = 42 hundreds ÷ 7
= 6 hundred
= 6 × 100
= 600
Hence, 4,200 ÷ 7 = 600

Question 13.
40 ÷ 2 = _____
Answer: 40 ÷ 2 = 20

Explanation:
By using the place-value method,
40 ÷ 2 = 4 tens ÷ 2
= 2 tens
= 2 × 10
= 20
Hence, 40 ÷ 2 = 20

Factors, Multiples, and Patterns Performance Task

You play basketball in a youth basketball program.

Question 1.
There are 72 players in the program. Each team needs an equal number of players and must have at least 5 players. What are two different ways the teams can be made?
Answer: The different ways the teams can be made and to have at least 5 members are: 8 × 9 and 9 × 8

Explanation:
Given that there are 72 players in the program and each team needs an equal number of players.
It is also given that each team must have at least 5 players.
Now,
To find the different ways the teams can be made, we have to find the factors of 72
Now,
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Now, to make an equal number of players and to have at least 5 players,
The different ways the teams can be made is: 8 × 9, 9 × 8
Hence, from the above,
We can conclude that the 2 different ways the teams can be made are: 8 × 9 and 9 × 8

Question 2.
The width of a basketball court is 42 feet and the length is 74 feet. You run around the perimeter of the court 4 times to warm up. How many feet do you run?

Answer: The number of feet you run is: 464 feet

Explanation:
Given that the width of a basketball court is 42 feet high and the length is 74 feet
So,
The perimeter of the basketball court = 42 + 74 = 116 feet
It is also given that you run around the perimeter of the basketball court 4 times to warm up.
So,
The number of feet you run to warm up = The perimeter of the basketball court × 4
= 116 × 4
Now,
By using the partial products method,
116 × 4 = ( 100 + 16 ) × 4
= ( 100 × 4 ) + ( 16 × 4 )
= 400 + 64
= 464 feet
Hence, from the above,
We can conclude that you will run up 464 feet to warm up your body.

Question 3.
You and your friend are on the same team. You played your first game last week.
a. Your team scored 4 more points than the other team. The total number of points scored by both teams was 58. How many points did your team score?
b. You and your friend scored the same number of points. You made 2-point shots and your friend made 3-point shots. What could be the greatest number of points you and your friend each scored?
Answer:

Question 4.
Your team uses the pattern below to decide which jersey color to wear to each game. Which color jersey will your team wear on the 20th game?

Answer: The color of jersey your team will wear on the 20th game is: Game 4

Explanation:
Given that your team uses the following pattern:

From the above pattern,
The total number of shirts is: 4
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
20 ÷ 4 = 5 R 0
Hence, from the above,
We can conclude that the color of the shirt your team will wear on the 20th game is: Game 4

Factors, Multiples, and Patterns Activity

Multiple Lineup
Directions:
1. Players take turns rolling a die.
2. On your turn, place a counter on a multiple of the number of your roll. If there is not a multiple of the number of your roll, you lose your turn.
3. The first player to create a line of 5 in a row, horizontally, vertically, or diagonally, wins!

Answer:

Factors, Multiples, and Patterns Chapter Practice

6.1 Understand Factors

Question 1.
Use the rectangles to find the factor pairs for 6.

Answer: The factor pairs of 6 are: 1 and 6, 2 and 3

Explanation:

The factor pairs are nothing but the side lengths of a rectangle and the area of a rectangle gives the factor
Hence,
The factor pairs of 6 are: 1 and 6, 2 and 3

Question 2.
Draw rectangles to find the factor pairs for 12.

Answer: The factor pairs of 12 are: 1 and 12, 2 and 6, 3 and 4

Explanation:

The factor pairs are nothing but the side lengths of a rectangle and the area of a rectangle gives the factor
Hence,
The factor pairs of 12 are: 1 and 12, 2 and 6, 3 and 4

Find the factor pairs for the number.
Question 3.
17
Answer: The factor pairs of 17 are: 1 and 17

Explanation:
Factors are the numbers that divide the original completely.
Hence,
The factor pairs of 17 are: 1 × 17

Question 4.
10
Answer: The factor pairs of 10 are: 1 and 10, 2 and 5

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 10 are: 1 × 10, 2 × 5

Question 5.
21
Answer: The factor pairs of 21 are: 1 and 21, 3 and 7

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 21 are: 1 × 21, 3 × 7

Question 6.
20
Answer: The factor pairs of 20 are: 1 and 20, 2 and 10, 4 and 5

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 20 are: 1 × 20, 2 × 10, 5 × 4

Question 7.
36
Answer: The factor pairs of 36 are: 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 36 are: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6

Question 8.
50
Answer: The factor pairs of 50 are: 1 and 50, 2 and 25, 5 and 10

Explanation:
factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 50 are: 1 × 50, 2 × 25, 5 × 10

6.2 Factors and Divisibility

Find the factor pairs for the number.
Question 9.
16
Answer: The factor pairs of 16 are: 1 and 16, 2 and 8, 4 and 4

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 16 are: 1 × 16, 2 × 8, 4 × 4

Question 10.
24
Answer: The factor pairs of 24 are: 1 and 24, 2 and 12, 3 and 8, 4 and 6

Explanation:
factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 24 are: 1 × 24, 2 × 12, 3 × 8, 4 × 6

Question 11.
56
Answer: The factor pairs of 56 are: 1 and 56, 2 and 28, 4 and 14, 7 and 8

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factor pairs of 56 are: 1 × 56, 2 × 28, 4 × 14, 7 × 8

List the factors of the number.
Question 12.
25
Answer: The factors of 25 are: 1, 5, 25

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factors of 25 are: 1, 5, 25

Question 13.
60
Answer: The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Question 14.
72
Answer: The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Explanation:
Factors are the numbers that divide the original number completely.
Hence,
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Question 15.
Number Sense
Which numbers have 3 as a factor?

Answer: The numbers which have 3 as a factor are: 21, 36, 48, 93

Explanation:
Given numbers are: 56, 21, 3, 48, 93, 71
For the given number to have 3 as a factor, the sum of the digits of the given number must be a multiple of 3
So,
The sum of digits of:
56: 5 + 6 = 11: 1 + 1 = 2
21: 2 + 1 = 3
36: 3 + 6 = 9
48: 4 + 8 = 12: 1 + 2 = 3
93: 9 + 3 = 12: 1 + 2 = 3
71: 7 + 1 = 8
Hence, from the above,
we can conclude that the numbers which have 3 as a factor are: 21, 36, 48, 93

6.3 Relate Factors and Multiples

Question 16.
Is 54 a multiple of 3? Explain.
Answer: 54 is a multiple of 3

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33
To be a multiple of 3, the sum of the digits of the given number should also be a multiple of 3.
So,
54: 5 + 4 = 9 ( Divisible by 3 )
Hence, from the above,
we can conclude that 54 is a multiple of 3

Question 17.
Is 45 a multiple of 7? Explain.
Answer: 45 is not a multiple of 7

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
The multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70
Hence, from the above,
we can conclude that 45 is not a multiple of 7

Question 18.
Is 2 a factor of 97? Explain.
Answer: 2 is not a factor of 97

Explanation:
Factors are the numbers that divide the original number completely.
The factors of 97 are: 1, 97
Hence, from the above,
We can conclude that 2 is not a factor of 97

Question 19.
Is 5 a factor of 60? Explain.
Answer: 5 is a factor of 60

Explanation:
Factors are the numbers that divide the original number completely.
The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Hence, from the above,
We can conclude that 5 is a factor of 60

Tell whether 20 is a multiple or a factor of the number. Write multiple, factor, or both.
Question 20.
60
Answer: 20 is a factor of 60

Explanation:
Factors are the numbers that divide the original number completely.
The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Hence, from the above,
We can conclude that 20 is a factor of 60

Question 21.
4
Answer: 20 is a multiple of 4

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Hence, from the above,
We can conclude that 20 is a multiple of 4

Question 22.
20
Answer: 20 is a multiple and factor of 20

Explanation:
Multiples are the numbers that can be divided by another number a certain number of times without any remainder.
Factors are the numbers that divide the original number completely
Now,
Factors of 20 are: 1, 2, 4, 5, 10, 20
Multiples of 20 are: 20, 40, 60, 80, 100
Hence, from the above,
we can conclude that 20 is a multiple and factor of 20

Question 23.
Number Sense
Name two numbers that are each a multiple of both 5 and 2. What do you notice about the two multiples?
Answer: The 2 numbers that are each a multiple of both 5 and 2 are: 10 and 20

Explanation;
Multiples of 2 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
So,
From the multiples of both 2 and 5,
The numbers which have common multiples of both 5 and 2 are: 10, 20
The multiples of 2 have the one’s digit as 2 or 4 or 6 or 8 or 0
The multiples of 5 have the one’s digit as 5 or 0

Question 24.
Logic
A quotient is a multiple of 5. The dividend is a multiple of 4. The divisor is a factor of 8. Write one possible equation for the problem.
Answer: The possible equation is:
80 ÷ 8 = 10

Explanation:
Given that,
Quotient: Multiple of 5
Dividend: Multiple of 4
Divisor: Factor of 8
Hence, from the above,
we can conclude that the possible equation for the problem is:
80 ÷ 8 = 10

6.4 Identify Prime and Composite Numbers

Tell whether the number is prime or composite. Explain.
Question 25.
5
Answer: 5 is a prime number

Explanation;
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Prime numbers”
Now,
The factors of 5 are: 1, 5
Hence, from the above,
We can conclude that 5 is a prime number

Question 26.
25
Answer: 25 is a composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Now,
The factors of 25 are: 1, 5, 25
Hence, from the above,
We can conclude that 25 is a composite number

Question 27.
51
Answer: 51 is a composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Now,
The factors of 51 are: 1, 3, 17, 51
Hence, from the above,
We can conclude that 51 is a composite number

Question 28.
21
Answer: 21 is a composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Now,
The factors of 21 are: 1, 3, 7, 21
Hence, from the above,
We can conclude that 21 is a composite number

Question 29.
50
Answer: 50 is a composite number

Explanation:
Composite numbers:
The numbers which have more than 2 factors are “Composite numbers”
Now,
The factors of 50 are: 1, 2, 5, 10, 25, 50
Hence, from the above,
We can conclude that 50 is a composite number

Question 30.
83
Answer: 83 is a prime number

Explanation:
Prime numbers:
The numbers which have exactly 2 factors 1 and itself are “Prime numbers”
Now,
The factors of 83 are: 1, 83
Hence, from the above,
We can conclude that 83 is a prime number

Question 31.
Modeling Real Life
A prime number of students have which type of fingerprint?

Answer: A prime number of students have Whorl fingerprint

Explanation:

From the given figure,
Each half-circle = 1 student
The number of fingerprints of loop-type fingerprint: 9
The number of fingerprints of Arch-type fingerprint: 12
The number of fingerprints on Whorl-type fingerprint: 7
Hence, from the above,
We can conclude that the prime number of students has Whorl-type fingerprint

6.5 Number Patterns

Write the first six numbers in the pattern. Then describe another feature of the pattern.
Question 32.
Rule: Subtract 11.
First number: 99
99, ___, _____, _____, _____, _____
Answer:
The first 6 numbers in the given pattern are: 99, 88, 77, 66, 55, 44

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Subtract 11
First number: 99
Hence,
The pattern we will obtain is:

Question 33.
Rule: Multiply by 5.
First number: 10
10, _____, _____, ____, _____, _____
Answer:
The first 6 numbers of the given pattern is: 10, 50, 250, 1,250, 6,250, 31,250

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Multiply by 5
First number: 10
Hence,
The pattern we will obtain is:

Question 34.
Rule: Add 8.
First number: 15
Answer:
The first 6 numbers of the given pattern are: 15, 23, 31, 39, 47, 55

Explanation:
For the formation of the pattern,
the rules are:
Rule: Add 8
First number: 15
Hence,
The pattern we will obtain is:

Question 35.
Rule: Divide by 4.
First number: 4,096
Answer:
The first 6 numbers in the given pattern is: 4,096, 1,024, 256, 64, 16, 4

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Divide by  4
First number: 4,096
Hence,
The pattern we will obtain is:

Open-Ended
Use the rule to generate a pattern of four numbers.
Question 36.
Rule: Divide by 2.
Answer:
Let the first number be 16
Hence,
The first 4 numbers of the given pattern are: 16, 8, 4, 2

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Divide by 2
Let the first number be 16
Hence,
The pattern we will obtain is:

Question 37.
Rule: Add 3.
Answer:
Let the first number be 3
Hence,
The first four numbers of the given pattern are: 3, 6, 9, 12

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Add 3
Let the first number be 3
Hence,
The pattern we will obtain is:

Question 38.
Rule: Multiply by10.
Answer:
Let the first number be 10
Hence,
The first  4 numbers of the given pattern are: 10, 100, 1,000, 10,000

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Multiply by 10
Let the first number be 10
Hence,
The pattern we will obtain is:

Question 39.
Rule: Subtract 6.
Answer:
Let the first number be 36
Hence,
The first 4 numbers of the given pattern are: 36, 30, 24, 18

Explanation:
For the formation of the pattern,
the given rules are:
Rule: Subtract 6
Let the first number be 36
Hence,
The pattern we will obtain is:

6.6 Shape Patterns

Question 40.
Extend the pattern of shapes by repeating the rule “trapezoid, circle.” What is the 57th shape in the pattern?

Answer: The 57th shape in the pattern is: Trapezoid

Explanation:
Given pattern is:

The rule given for the pattern is: trapezoid, circle
So,
The total number of figures in the given pattern is: 2
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotient method,
57 ÷ 2 = ( 40 + 16 ) ÷ 2
= ( 40 ÷ 2 ) + ( 16 ÷ 2 )
= 20 + 8
= 28 R 1
Hence, from the above,
we can conclude that the 57th shape in the pattern is: Trapezoid

Question 41.
Extend the pattern of shapes by repeating the rule “top left, top right, bottom right, bottom left.” What is the 102nd shape in the pattern?

Answer: The 102nd shape in the pattern is: Bottom left

Explanation:
The given pattern is:

The rule given for the pattern is: top left, top right, bottom right, bottom left
So,
The total number of figures in the given pattern is: 4
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
102 ÷ 3 = ( 99 + 3 ) ÷ 3
= ( 99 ÷ 3 ) + ( 3 ÷ 3 )
= 33 + 1
= 34 R 0
Hence, from the above,
We can conclude that the 102nd shape in the pattern is: Bottom left

Question 42.
Describe the pattern. How many squares are in the 61st figure?

Answer: The number of squares in the 61st figure is: 3

Explanation:
The given pattern is:

In figure 1, the total number of squares = 3
In figure 2, the total number of squares = 6
In figure 3, the total number of squares = 9
So,
The total number of figures = 3
So,
for this, when we divide any number by the total number of patterns, the value of the quotient will be a total repetition of the patterns and the remainder will be the counting of the pattern from the 1st figure.
So,
By using the partial quotients method,
61 ÷ 3 = ( 57 + 3 ) ÷ 3
= ( 57 ÷ 3 ) + ( 3 ÷ 3 )
= 19 + 1
= 20 R 1
Hence, from the above,
We can conclude that the number of squares in the  61st figure in the pattern is: 4

Question 43.
Structure
Draw the missing figure in the pattern. Explain the pattern.

Answer: The missing figure has: 7 squares

Explanation:
The given pattern is:

The rule-following in the given pattern is: Add 2
The first number: 1
According to the rule,
The number of squares in the pattern will be like 1, 3, 5, 7, 9

Conclusion:

We hope the details mentioned in the Big Ideas Math Answers Grade 4 Chapter 6 Factors, Multiples, and Patterns are helpful for you to finish your homework in time. Learn the concepts in depth so that you can solve any kind of problem easily. If you have any queries you can post your comments in the below section.