Lesson 8
Moves in Parallel
Problem 1
- Draw parallel lines \(AB\) and \(CD\).
- Pick any point \(E\). Rotate \(AB\) 90 degrees clockwise around \(E\).
- Rotate line \(CD\) 90 degrees clockwise around \(E\).
- What do you notice?
Solution
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Problem 2
Use the diagram to find the measures of each angle. Explain your reasoning.
- \(m{\angle ABC}\)
- \(m{\angle EBD}\)
- \(m{\angle ABE}\)
![Lines A D and E C intersect at point B. Angle C B D is 50 degrees.](https://cms-im.s3.amazonaws.com/DRwwr7b3tnu8NSAFof8iopuL?response-content-disposition=inline%3B%20filename%3D%228-8.1.B9.newPP.01.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.B9.newPP.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T202659Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=96ebaaf8ae64a8f8f364bbda957a86de58e61ee5080e7b2706a3a10218b47ac9)
Solution
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Problem 3
Points \(P\) and \(Q\) are plotted on a line.
![A line that slants upward and to the right with two plots labeled P and Q pointed on it. Point P is below point Q.](https://cms-im.s3.amazonaws.com/eVe5dJur4Z24KoZB4fsjyXJB?response-content-disposition=inline%3B%20filename%3D%228-8.1.B.PP.Image.12.5.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.B.PP.Image.12.5.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T202659Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8caf07681a7f8f4ec19ae9d804935c25ed8e190b0a75f4e9955e86357b4c9db0)
- Find a point \(R\) so that a 180-degree rotation with center \(R\) sends \(P\) to \(Q\) and \(Q\) to \(P\).
- Is there more than one point \(R\) that works for part a?
Solution
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Problem 4
In the picture triangle \(A’B’C’\) is an image of triangle \(ABC\) after a rotation. The center of rotation is \(D\).
![A triangle A B C and its image, triangle A prime B prime C prime and a point D. Side B C is 4, angle C prime is 50 degrees and angle B prime is 52 degrees.](https://cms-im.s3.amazonaws.com/373faDYwrb8nv4ubTzSyjFVt?response-content-disposition=inline%3B%20filename%3D%228-8.1.B.PP.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.B.PP.Image.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T202659Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a2042e3c8f4cd3f0f159e4d81e83e3b71af4b2da41fc38fe4949fdf071404acb)
- What is the length of side \(B’C’\)? Explain how you know.
- What is the measure of angle \(B\)? Explain how you know.
- What is the measure of angle \(C\)? Explain how you know.
Solution
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(From Unit 1, Lesson 6.)Problem 5
The point \((\text-4,1)\) is rotated 180 degrees counterclockwise using center \((0,0)\). What are the coordinates of the image?
\((\text-1,\text-4)\)
\((\text-1,4)\)
\((4,1)\)
\((4,\text-1)\)
Solution
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(From Unit 1, Lesson 5.)