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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T160028Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ecea0a0f4015c79f217abae1e1b32804b297420e9270f9054b30f0c76bb37690)
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T160028Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=897670bebaa3583dacabd1d94ca15ca00b09fc4fb28f97ec2f479a0c462fc629)
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T160028Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=89055aa1be8bae3f5c8bf6af7d19ed3c52236d9438c2afd71e796139f4b1d49d)
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T160028Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8ad91df75eabcf840b1ca653d42b075deba0da7b250379365cc58717ae3b69b4)
- 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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T160028Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=fd1e6e1f7cdf7131c3358b69e48f5fb2864b859e4311a8b414ccb7af0c6b7643)
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T160028Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c1812c2585453c093043bd791ac16e8f28335f7aaa750729c2ec0fc848ffe5f8)