![\"Angle](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/03\/angle-between-two-straight-lines.png\")
<\/p>\n
Two straight lines L\u2081, L\u2082 are intersecting each other to form acute and obtuse angles.<\/p>\n
Let us take the slope measurement can be taken as<\/p>\n
tan \u03b8\u2081 = m\u2081 and tan \u03b8\u2082 = m\u2082<\/p>\n
From the figure, we can say that \u03b8 = \u03b8\u2082 – \u03b8\u2081<\/p>\n
Now, tan \u03b8 = tan(\u03b8\u2082 – \u03b8\u2081)<\/p>\n
tan \u03b8 = \\(\\frac { (tan \u03b8\u2082 – tan \u03b8\u2081) }{ (1 + tan \u03b8\u2081 tan \u03b8\u2082) } \\)<\/p>\n
Substitute tan \u03b8\u2081 = m\u2081, tan \u03b8\u2082 = m\u2082<\/p>\n
tan \u03b8 = \\(\\frac { (m\u2082 – m\u2081) }{ (1 + m\u2081 m\u2082) } \\)<\/p>\n
How to find Angle Between Two Straight Lines?<\/h3>\n
If three points on a coordinate plane are given, then endpoints of a line are (x\u2081, y\u2081) and (x\u2082, y\u2082)<\/p>\n
The equation of the slope is m = \\(\\frac { (y\u2082 – y\u2081) }{ (x\u2082 – x\u2081) } \\)<\/p>\n
m\u2081 and m\u2082 can be calculated by substituting in the formula, then the angle between two lines is given by<\/p>\n
tan \u03b8 = \u00b1 \\(\\frac { m\u2082 – m\u2081) }{ (1 + m\u2081 m\u2082) } \\)<\/p>\n
Also Check:<\/p>\n
- \n
- Pairs of Angles<\/a><\/li>\n
- Angle of Elevation<\/a><\/li>\n
- Angle of Depression<\/a><\/li>\n<\/ul>\n
Angle between Two Straight Lines Examples<\/h3>\n
Example 1:<\/strong><\/p>\n
If A (-2, 1), B (2, 3), and C (-2, -4) are three points, find the angle between two straight lines AB, BC.<\/p>\n
Solution:<\/strong><\/p>\n
Given that,<\/p>\n
Three points are A (-2, 1), B (2, 3), and C (-2, -4)<\/p>\n
The slope of line AB is m = \\(\\frac { (y\u2082 – y\u2081) }{ (x\u2082 – x\u2081) } \\)<\/p>\n
m = \\(\\frac { (3 – 1) }{ (2 – (-2)) } \\)<\/p>\n
= \\(\\frac { (2) }{ (2 + 2) } \\)<\/p>\n
= \\(\\frac { 2 }{ 4 } \\)<\/p>\n
= \\(\\frac { 1 }{ 2 } \\)<\/p>\n
Therefore, m\u2081 = \\(\\frac { 1 }{ 2 } \\)<\/p>\n
The slope of line BC is given by<\/p>\n
m = \\(\\frac { (y\u2082 – y\u2081) }{ (x\u2082 – x\u2081) } \\)<\/p>\n
m = \\(\\frac { (-4 – 3) }{ (-2 – 2) } \\)<\/p>\n
= \\(\\frac { -7 }{ -4 } \\)<\/p>\n
= \\(\\frac { 7 }{ 4 } \\)<\/p>\n
Therefore, m\u2082 = \\(\\frac { 7 }{ 4 } \\)<\/p>\n
Substituting the values of m2 and m1 in the formula for the angle between two lines when we know the slopes of two sides, we have,<\/p>\n
tan \u03b8 = \u00b1 \\(\\frac { (m\u2082 – m\u2081) }{ (1 + m\u2081 m\u2082) } \\)<\/p>\n
= \u00b1 \\(\\frac { (\\frac { 7 }{ 4 } – \\frac { 1 }{ 2 } ) }{ (1 + \\frac { 1 }{ 2 }\u00a0 * \\frac { 7 }{ 4 }) } \\)<\/p>\n
= \u00b1 \\(\\frac { 2 }{ 3 } \\)<\/p>\n
Therefore, \u00a0\u03b8 = tan\u00a0-1<\/sup>\u00a0(\u2154)<\/p>\n
So, the angle between the lines AB, BC is tan\u00a0-1<\/sup> (\u2154).<\/p>\n
Example 2:<\/strong><\/p>\n
Find the angle between the following lines 4x – 3y = 8, 2x + 5y = 4.<\/p>\n
Solution:<\/strong><\/p>\n
Given two straight lines are 4x – 3y = 8, 2x + 5y = 4<\/p>\n
Converting the given lines into slope intercept form<\/p>\n
4x – 3y = 8<\/p>\n
4x = 8 + 3y<\/p>\n
3y = 4x – 8<\/p>\n
y = \\(\\frac { 4x – 8 }{ 3 } \\)<\/p>\n
y = \\(\\frac { 4x }{ 3 } – \\frac { 8 }{ 3 } \\)<\/p>\n
Therefore, the slope of line 4x – 3y = 8 is \\(\\frac { 4 }{ 3 } \\)<\/p>\n
2x + 5y = 4<\/p>\n
5y = 4 – 2x<\/p>\n
y = \\(\\frac { 4 – 2x }{ 5 } \\)<\/p>\n
y = \\(\\frac { -2x }{ 5 } + \\frac { 4 }{ 5 } \\)<\/p>\n
Therefore, the slope of the line 2x + 5y = 4 is –\\(\\frac { 2 }{ 5 } \\)<\/p>\n
The angle between lines is tan \u03b8 = \u00b1 \\(\\frac { (m\u2082 – m\u2081) }{ (1 + m\u2081 m\u2082) } \\)<\/p>\n
= \u00b1 \\(\\frac { (\\frac { -2 }{ 5 } – \\frac { 4 }{ 3 } ) }{ (1 + \\frac { 4 }{ 3 }\u00a0 * \\frac { (-2) }{ 5 }) } \\)<\/p>\n
= \\(\\frac { -26 }{ 7 } \\)<\/p>\n
\u03b8 = tan\u00a0-1 <\/sup> (\\(\\frac { -26 }{ 7 } \\))<\/p>\n
Example 3:<\/strong><\/p>\n
Find the angle between two lines x + y = 4, x + 2y = 3.<\/p>\n
Solution:<\/strong><\/p>\n
The given two lines are x + y = 4, x + 2y = 3.<\/p>\n
The slope-intercept form of the first line is<\/p>\n
x + y = 4<\/p>\n
y = 4 – x<\/p>\n
Therefore, slope of x + y = 4 is m\u2081 = -1<\/p>\n
The slope-intercept form of the second line is<\/p>\n
x + 2y = 3<\/p>\n
2y = 3 – x<\/p>\n
y = \\(\\frac { (3 – x) }{ 2 } \\)<\/p>\n
y = \\(\\frac { 3 }{ 2 } – \\frac { x }{ 2 } \\)<\/p>\n
Therefore, slope of x + 2y = 3 is m\u2082 = \\(\\frac { -1 }{ 2 } \\)<\/p>\n
The angle between lines is tan \u03b8 = \u00b1 \\(\\frac { (m\u2082 – m\u2081) }{ (1 + m\u2081 m\u2082) } \\)<\/p>\n
= \u00b1 \\(\\frac { (\\frac { -1 }{ 2 } – (-1) ) }{ (1 + \\frac { -1 }{ 2 }\u00a0 * (-1)) } \\)<\/p>\n
= \\(\\frac { 1 }{ 3 } \\)<\/p>\n
\u03b8 = tan-1<\/sup>(\\(\\frac { 1 }{ 3 } \\))<\/p>\n
Example 4:<\/strong><\/p>\n
Find the angle between two straight lines x + 2y – 1 = 0 and 3x – 2y + 5 = 0<\/p>\n
Solution:<\/strong><\/p>\n
The given lines are x + 2y – 1 = 0 and 3x – 2y + 5 = 0<\/p>\n
The slope intercept form of first line is<\/p>\n
x + 2y – 1 = 0<\/p>\n
2y = 1 – x<\/p>\n
y = \\(\\frac { 1 – x }{ 2 } \\)<\/p>\n
y = \\(\\frac { 1 }{ 2 } – \\frac { x }{ 2 } \\)<\/p>\n
Therefore, the slope of line x + 2y – 1 = 0 is m\u2081 = \\(\\frac { -1 }{ 2 } \\)<\/p>\n
The slope-intercept form of the second line is<\/p>\n
3x – 2y + 5 = 0<\/p>\n
3x + 5 = 2y<\/p>\n
y = \\(\\frac { 3x + 5 }{ 2 } \\)<\/p>\n
y = \\(\\frac { 3x }{ 2 } + \\frac { 5 }{ 2 } \\)<\/p>\n
Therefore, the slope of line 3x – 2y + 5 = 0 is m\u2082 = \\(\\frac { 3 }{ 2 } \\)<\/p>\n
The angle between lines is tan \u03b8 = \u00b1 \\(\\frac { (m\u2082 – m\u2081) }{ (1 + m\u2081 m\u2082) } \\)<\/p>\n
= \u00b1 \\(\\frac { (\\frac { 3 }{ 2 } – \\frac { (-1 }{ 2 )} ) }{ (1 – \\frac { 1 }{ 2 }\u00a0 * \\frac { (3) }{ 2 }) } \\)<\/p>\n
= 2<\/p>\n
\u03b8 = tan-1<\/sup>(2)<\/p>\n","protected":false},"excerpt":{"rendered":"
The angle is nothing but the figure formed by two rays. If two straight lines meet, then they form two sets of angles. The intersection forms a pair of acute angles and another pair of obtuse angles. The angle values will be based on the slopes of the intersecting lines. Check out the formula to … Read more<\/a><\/p>\n","protected":false},"author":12,"featured_media":60707,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[],"jetpack_featured_media_url":"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/03\/Angle-between-Two-Straight-Lines-1.png","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/56479"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=56479"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/56479\/revisions"}],"predecessor-version":[{"id":92362,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/56479\/revisions\/92362"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media\/60707"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=56479"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=56479"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=56479"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- Angle of Elevation<\/a><\/li>\n