Lesson 13
Similar Triangles
Let’s look at similar triangles.
Problem 1
In each pair, some of the angles of two triangles in degrees are given. Use the information to decide if the triangles are similar or not. Explain how you know.
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Triangle A: 53, 71, ___; Triangle B: 53, 71, ___
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Triangle C: 90, 37, ___; Triangle D: 90, 53, ___
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Triangle E: 63, 45, ____; Triangle F: 14, 71, ____
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Triangle G: 121, ___, ___; Triangle H: 70, ___, ___
Problem 2
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Draw two equilateral triangles that are not congruent.
- Measure the side lengths and angles of your triangles. Are the two triangles similar?
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Do you think two equilateral triangles will be similar always, sometimes, or never? Explain your reasoning.
Problem 3
In the figure, line \(BC\) is parallel to line \(DE\).
![Two triangles. First, A, B C. Then triangle A, D E where D lies on side A, B and E lies on side A, E.](https://staging-cms-im.s3.amazonaws.com/Qhexj1cMRXNhDkvRaax6tphz?response-content-disposition=inline%3B%20filename%3D%228-8.2.B.PP.Image.13.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B.PP.Image.13.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T083231Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b7abbe89f28e28542f3220ad0eb29babb1a6ffa421cd696890233a9710ce7cd3)
Explain why \(\triangle ABC\) is similar to \(\triangle ADE\).
Problem 4
The quadrilateral \(PQRS\) in the diagram is a parallelogram. Let \(P’Q’R’S’\) be the image of \(PQRS\) after applying a dilation centered at a point O (not shown) with scale factor 3.
![Quadrilateral P Q R S on a square grid.](https://staging-cms-im.s3.amazonaws.com/wmd3NdUusS4hd9qSy5RoCN8b?response-content-disposition=inline%3B%20filename%3D%228-8.2.A.PP.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.A.PP.Image.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T083231Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=598d88af0975839d9aa355272ca44dfc55b007e7b2bcbb22b69b04ba3c0d10f1)
Which of the following is true?
\(P’Q’= PQ\)
\(P’Q’=3PQ\)
\(PQ=3P’Q’\)
Cannot be determined from the information given
Problem 5
Describe a sequence of transformations for which Quadrilateral P is the image of Quadrilateral Q.
![Two quadrilaterals in an x y plane.](https://staging-cms-im.s3.amazonaws.com/qBPDuyfnrmYr6yyg3hYGfSZq?response-content-disposition=inline%3B%20filename%3D%228-8.1.A6.newPP.03b.png%22%3B%20filename%2A%3DUTF-8%27%278-8.1.A6.newPP.03b.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T083231Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e798457bfc35d5a2ee1462fca70ac222ad20ced0b54cdb389c7bfa325b034a0f)
Problem 6
Polygon B is a scaled copy of Polygon A.
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What is the scale factor from Polygon A to Polygon B? Explain your reasoning.
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Find the missing length of each side marked with ? in Polygon B.
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Determine the measure of each angle marked with ? in Polygon A.
![2 pentagons, labeled A and B.](https://staging-cms-im.s3.amazonaws.com/9R95g8zhh8RD5ZhFiBCQpYjW?response-content-disposition=inline%3B%20filename%3D%227-7.1.A.PP.Image.18.png%22%3B%20filename%2A%3DUTF-8%27%277-7.1.A.PP.Image.18.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T083231Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=be222ef3113b5c908dc1129f0a110207dee47be55a62365f50a40dbfa8545a29)