6.5 – Prove Triangles Similar by SSS and SAS

1. 6.5 – Prove Triangles Similar by SSS and SAS Geometry Ms. Rinaldi

2. Side-Side-Side (SSS) Similarity Theorem If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

3. = = = = CA 4 4 8 16 4 AB 12 BC Is either DEF or GHJsimilar to ABC? FD 9 DE 3 3 EF 3 12 6 All of the ratios are equal, so ABC~DEF. ANSWER Compare ABCand DEFby finding ratios of corresponding side lengths. = = EXAMPLE 1 Use the SSS Similarity Theorem SOLUTION Remaining sides Shortest sides Longest sides

4. 1 = = = = 8 CA 16 BC 12 6 AB The ratios are not all equal, so ABCand GHJare not similar. HJ 8 JG 10 GH 5 16 ANSWER Compare ABCand GHJby finding ratios of corresponding side lengths. 1 = = EXAMPLE 1 Use the SSS Similarity Theorem (continued) Remaining sides Longest sides Shortest sides

5. Which of the three triangles are similar? Write a similarity statement. Use the SSS Similarity Theorem EXAMPLE 2

6. ALGEBRA Find the value of xthat makes ABC ~ DEF. 4 x–1 4 18 = 12(x – 1) 12 18 STEP1 Find the value of xthat makes corresponding side lengths proportional. = EXAMPLE 3 Use the SSS Similarity Theorem SOLUTION Write proportion. Cross Products Property 72 = 12x – 12 Simplify. 7 = x Solve for x.

7. ? = = ANSWER AC 6 4 BC 8 4 AB AB STEP2 Check that the side lengths are proportional when x = 7. 24 12 12 18 DE EF DE DF When x = 7, the triangles are similar by the SSS Similarity Theorem. ? = = EXAMPLE 3 Use the SSS Similarity Theorem (continued) DF = 3(x + 1) = 24 BC = x – 1 = 6

8. Use the SSS Similarity Theorem EXAMPLE 4 Find the value of x that makes Q Y 20 30 x + 6 21 X Z 12 P R 3(x – 2)

9. Side-Angle-Side (SAS) Similarity Theorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

10. Lean-to Shelter You are building a lean-to shelter starting from a tree branch, as shown. Can you construct the right end so it is similar to the left end using the angle measure and lengths shown? EXAMPLE 5 Use the SAS Similarity Theorem

11. ANSWER Both m A andm F equal = 53°, so A F. Next, compare the ratios of the lengths of the sides that include A and F. So, by the SAS Similarity Theorem, ABC~FGH. Yes, you can make the right end similar to the left end of the shelter. The lengths of the sides that include Aand F are proportional. 15 AB 3 3 9 AC = = FG 2 6 10 2 FH ~ = = EXAMPLE 5 Use the SAS Similarity Theorem (continued) SOLUTION Shorter sides Longer sides

12. 18 9 3 CA 3 BC 5 CD 30 5 15 EC = = = = The corresponding side lengths are proportional. The included angles ACB and DCEare congruent because they are vertical angles. So, ACB ~DCE by the SAS Similarity Theorem. EXAMPLE 6 Choose a method Tell what method you would use to show that the triangles are similar. SOLUTION Find the ratios of the lengths of the corresponding sides. Shorter sides Longer sides

13. Explain how to show that the indicated triangles are similar. Explain how to show that the indicated triangles are similar. B. XZW ~ YZX A. SRT ~ PNQ Choose a method EXAMPLE 7