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Are the triangles similar? If so, which similarity theorem is used?

Are the triangles similar? If so, which similarity theorem is used?

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Are the triangles similar? If so, which similarity theorem is used?** Two triangles are shown side by side: **Triangle STU:** - Side ST = 5 - Side SU = 4 - Angle ∠S = 57° **Triangle XYZ:** - Side XY = 15 - Side XZ = 12 - Angle ∠X = 57° **Multiple Choice Answers:** - O Yes, ΔSTU ~ ΔXYZ by the SAS Similarity Theorem. - O Yes, ΔSTU ~ ΔXYZ by the AA Similarity Theorem. - O No, the triangles are not similar. - O Yes, ΔSTU ~ ΔXYZ by the SSS Similarity Theorem. The problem is to determine whether Triangle STU and Triangle XYZ are similar, and if they are, to identify the theorem used to establish their similarity. ### Explanation: **Graph or Diagram Explanation:** The two triangles shown in the diagram both have one angle measuring 57°. The corresponding side lengths adjacent to the 57° angles are proportionate (5/15 = 1/3 and 4/12 = 1/3). ### Concepts Used: 1. **SAS (Side-Angle-Side) Similarity Theorem:** Two triangles are similar if two sides in one triangle are in the same proportion as two sides in the other triangle, and the included angle between those sides are equal. 2. **AA (Angle-Angle) Similarity Theorem:** Two triangles are similar if any two corresponding angles are equal. 3. **SSS (Side-Side-Side) Similarity Theorem:** Two triangles are similar if all three corresponding sides are in proportion. ### Solution: Since both triangles have one angle measuring 57° and the sides adjacent to these angles are in proportion, the triangles are similar by the SAS Similarity Theorem. Thus, the correct answer is: **Yes, ΔSTU ~ ΔXYZ by the SAS Similarity Theorem.**
Transcribed Image Text:**Are the triangles similar? If so, which similarity theorem is used?** Two triangles are shown side by side: **Triangle STU:** - Side ST = 5 - Side SU = 4 - Angle ∠S = 57° **Triangle XYZ:** - Side XY = 15 - Side XZ = 12 - Angle ∠X = 57° **Multiple Choice Answers:** - O Yes, ΔSTU ~ ΔXYZ by the SAS Similarity Theorem. - O Yes, ΔSTU ~ ΔXYZ by the AA Similarity Theorem. - O No, the triangles are not similar. - O Yes, ΔSTU ~ ΔXYZ by the SSS Similarity Theorem. The problem is to determine whether Triangle STU and Triangle XYZ are similar, and if they are, to identify the theorem used to establish their similarity. ### Explanation: **Graph or Diagram Explanation:** The two triangles shown in the diagram both have one angle measuring 57°. The corresponding side lengths adjacent to the 57° angles are proportionate (5/15 = 1/3 and 4/12 = 1/3). ### Concepts Used: 1. **SAS (Side-Angle-Side) Similarity Theorem:** Two triangles are similar if two sides in one triangle are in the same proportion as two sides in the other triangle, and the included angle between those sides are equal. 2. **AA (Angle-Angle) Similarity Theorem:** Two triangles are similar if any two corresponding angles are equal. 3. **SSS (Side-Side-Side) Similarity Theorem:** Two triangles are similar if all three corresponding sides are in proportion. ### Solution: Since both triangles have one angle measuring 57° and the sides adjacent to these angles are in proportion, the triangles are similar by the SAS Similarity Theorem. Thus, the correct answer is: **Yes, ΔSTU ~ ΔXYZ by the SAS Similarity Theorem.**
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