Lesson 14
Alternate Interior Angles
Let’s explore why some angles are always equal.
Problem 1
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=20240722T160129Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c0b3af81af69a1f23b8a99a4fbc7aea1cf84058c169a953b6bda5fa423768d5f)
Problem 2
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=20240722T160129Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5ee486eacea62575ea39c645a65b9c555b0f01922903be7c6a831e1c0e46905b)
- 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 3
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=20240722T160129Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ddb2514201b0d45df16483bb3364708e5e4831e3bd8ad1434ebfaa1728644bd0)
Find the missing angle measures marked with question marks.
Problem 4
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=20240722T160129Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=826153a6bef52a172ca9fa9bbd949495f0714623931c79fa4816f6a0443fb540)
Problem 5
The two figures are scaled copies of each other.
- What is the scale factor that takes Figure 1 to Figure 2?
- What is the scale factor that takes Figure 2 to Figure 1?
![Two identical quadrilaterals on a grid.](https://staging-cms-im.s3.amazonaws.com/2HiYuiLWKwCpKZp1muKQeR6s?response-content-disposition=inline%3B%20filename%3D%228-8.1.PP.7Grev6.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.PP.7Grev6.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=20240722T160129Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7be89e376279257265574316a318a4d2db99bd104f77cad2aa96f887fc80348e)