Lesson 12
Alternate Interior Angles
Let’s explore why some angles are always equal.
Problem 1
Segments \(AB\), \(EF\), and \(CD\) intersect at point \(C\), and angle \(ACD\) is a right angle. Find the value of \(g\).
![Segment A, B, segment E F, and segment C D intersect at point C. Clockwise, the endpoints are A, D, E, B, F. Angle A, C D is a right angle. Angle D C E is 53 degrees, angle E C B is g degrees.](https://cms-im.s3.amazonaws.com/LQ3RKdH1ETNirtxGBnkUgwCF?response-content-disposition=inline%3B%20filename%3D%227-7.7.A4.new.PP.06.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.A4.new.PP.06.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T033855Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=cf2eb0b946f313caa429e40b0cac745cb5b082f8383c5815109008d9ea537c32)
Problem 2
\(M\) is a point on line segment \(KL\). \(NM\) is a line segment. Select all the equations that represent the relationship between the measures of the angles in the figure.
![M is a point on line segment K L. Segment N M creates two angles, measure a, degrees and b degrees.](https://cms-im.s3.amazonaws.com/dKpGbjgcTi5zugiPN5WT8z7R?response-content-disposition=inline%3B%20filename%3D%227-7.7.4.new.PP.01.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.4.new.PP.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T033855Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c465501ad08e943366802960724e20884cf93c998142b70014476b4168b91b92)
A:
\(a=b\)
B:
\(a+b=90\)
C:
\(b=90-a\)
D:
\(a+b=180\)
E:
\(180-a=b\)
F:
\(180=b-a\)
Problem 3
Use the diagram to find the measure of each angle.
- \(m\angle ABC\)
- \(m\angle EBD\)
- \(m\angle ABE\)
![Two lines, line E C and line A D, that intersect at point B. Angle C B D is labeled 45 degrees.](https://cms-im.s3.amazonaws.com/5ZyMLomGE6YoMm3pmEMVXaaS?response-content-disposition=inline%3B%20filename%3D%228.1.D.PP.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%278.1.D.PP.Image.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T033855Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5ead54feac2867170050b7335172d60fb29e11394f3866814598812d7fabf96d)
Problem 4
Lines \(k\) and \(\ell\) are parallel, and the measure of angle \(ABC\) is 19 degrees.
![Two parallel lines, k and l, cut by transversal line m.](https://cms-im.s3.amazonaws.com/DEJUC5fkJ5jfMnzU5gjdQA85?response-content-disposition=inline%3B%20filename%3D%228-8.1.B.PP.Image.12.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.B.PP.Image.12.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T033855Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=034331874eb83a06517d9c458aa4a8a8b8b270f46ed787cb6fef890c9e19e178)
- Explain why the measure of angle \(ECF\) is 19 degrees. If you get stuck, consider translating line \(\ell\) by moving \(B\) to \(C\).
- What is the measure of angle \(BCD\)? Explain.
Problem 5
The diagram shows three lines with some marked angle measures.
![Two lines that do not intersect. A third line intersects with both lines.](https://cms-im.s3.amazonaws.com/23cy1uwuxU8NDj2Eb5D4jMR8?response-content-disposition=inline%3B%20filename%3D%228-8.1.D14.newPP.01.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.D14.newPP.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T033855Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ca6c0d30c5b3dd0b5f0aa8c35c049252b733377d15997b7d31a3e51f53308ddb)
Find the missing angle measures marked with question marks.
Problem 6
Lines \(s\) and \(t\) are parallel. Find the value of \(x\).
![Four lines. Two parallel lines are labeled s and t. Two other lines that intersect at a right angle at a point on line t. One angle is labeled 40 degrees. Another angle is labeled x degrees.](https://cms-im.s3.amazonaws.com/C7VehJtgLrE7Ymp34nvB1iAG?response-content-disposition=inline%3B%20filename%3D%22angle%20diagram.png%22%3B%20filename%2A%3DUTF-8%27%27angle%2520diagram.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240718T033855Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4e0738dbbd427e7c07a99b069a2c02fc3659121f0d414f5aafd4a96b81d9f801)