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Understanding Triangle Similarity: Proving AA, SSS, and SAS

This lesson explores the concept of triangle similarity, focusing on the Angle-Angle (AA), Side-Side-Side (SSS), and Side-Angle-Side (SAS) criteria for proving triangles are similar. Through engaging examples, students will learn how to identify similar triangles, write similarity statements, and apply theorems in problem-solving. Topics include using vertical angles, right angle congruence, and properties of similar triangles. The lesson is designed to enhance understanding and application of geometric principles in various scenarios.

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Understanding Triangle Similarity: Proving AA, SSS, and SAS

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  1. Triangle Similarity: AA, SSS, and SAS 7-3 Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. Objectives Prove certain triangles are similar by using AA, SSS, and SAS. Use triangle similarity to solve problems.

  3. There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.

  4. Example 1 Explain why the triangles are similar and write a similarity statement. BCA  ECD Vertical Angles A  D Right Angle Congruence Theorem. Therefore ∆ABC ~ ∆DEC by AA~.

  5. Example 2 Explain why the triangles are similar and write a similarity statement. mC = 47°, so C F B E Right Angle Congruence Theorem Therefore, ∆ABC ~ ∆DEF by AA ~.

  6. Example 3 Verify that the triangles are similar. ∆PQR and ∆STU Therefore ∆PQR ~ ∆STU by SSS ~.

  7. Example 4 Verify that the triangles are similar. ∆DEF and ∆HJK D  H Therefore ∆DEF ~ ∆HJK by SAS ~.

  8. Example 5 Verify that ∆TXU ~ ∆VXW. TXU VXW Vertical Angles Theorem. Therefore ∆TXU ~ ∆VXW by SAS ~.

  9. Example 6 Explain why ∆ABE ~ ∆ACD, and then find CD. Step 1 Prove triangles are similar. A A by Reflexive Property of , and B C since they are both right angles. Therefore ∆ABE ~ ∆ACD by AA ~.

  10. Example 6 Step 2 Find CD. x(9) = 5(12) 9x = 60

  11. Example 7 Explain why ∆RSV ~ ∆RTU and then find RT. Step 1 Prove triangles are similar. It is given that S T. R R by Reflexive Property of . Therefore ∆RSV ~ ∆RTU by AA ~.

  12. Example 7 Step 2 Find RT. RT(8) = 10(12) 8RT = 120 RT = 15

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