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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T010331Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3bc38bfdacaf80f967aa4cf6562da56f1699d184cb91de1e38df4d9bba933a6f)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T010331Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=249e84699c4de6d00a1e61afb75bc908583a193645b8e028bf18cf4de8991a53)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T010331Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f4d3a523f2a605def54982b7a8a6f3d599e1ca230c94ad6c95c48054ddb62fea)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T010331Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d61c05cd1a650b37d8d6c03f5d401d4568e7779adc055c0efa8ddc6b02f50370)
- 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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T010331Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ac8af7ad0695d80a7f00b53d67e76bb3c7c2d0c5a361b473680afda84051b7c0)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T010331Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=54ec92e4fbb26b2305e13e2f3eba166b807f080b4da4dcb4044dd49d93da99cb)