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://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=20240727T010227Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f1902c6aade9ee429279fa5899d6e5d67782aab643ee7b211480c2ae8d1d9b53)
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://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=20240727T010227Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=22db7835908d78d0cbf184cd130700b1924a6312a2b3e43fe75c7b1f19153561)
- 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://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=20240727T010227Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=22ac4aa641c92ffae1237f0b9ae474e97e97ab2a83669364961d8ba36c06836a)
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://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=20240727T010227Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=448a4a2b35c295c28d0cec30b896253679049f039cb1a313c698a01f3c2d4292)
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://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=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T010227Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8c5320e589d60c0f5db3171b3e85d0468f04f98d3ac712df66de94ab47e18cf4)