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LESSON 8.5 NAME _________________________________________________________ DATE ____________ Practice with Examples For use with pages 488–496 GOAL Use similarity theorems to prove that two triangles are similar VOCABULARY Theorem 8.2 Side-Side-Side (SSS) Similarity Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. Theorem 8.3 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. EXAMPLE 1 Using the SSS Similarity Theorem Which of the following triangles are similar? A 3 B E 4 C 4 G 4 3 10 D 5 F 6 J 8 H SOLUTION To decide which, if any, of the triangles are similar, you need to consider the ratios of the lengths of corresponding sides. Ratios of Side Lengths of ABC and DEF CA 3 DF 5 Longest sides AB 2 1 EF 4 2 Remaining sides Chapter 8 BC 1 DE 3 Shortest sides Because the ratios are not equal, ABC and DEF are not similar. Ratios of Side Lengths of GHJ and DEF GH 6 2 DE 3 1 GJ 10 2 DF 5 1 8 2 HJ EF 4 1 Shortest sides Longest sides Remaining sides Because the ratios are equal, GHJ ~ DEF. Since DEF is similar to GHJ and DEF is not similar to ABC, GHJ is not similar to ABC. Copyright © McDougal Littell Inc. All rights reserved. Geometry Practice Workbook with Examples 151 LESSON 8.5 CONTINUED NAME _________________________________________________________ DATE ____________ Reteaching with Practice For use with pages 488–496 Exercises for Example 1 Determine which two of the three given triangles are similar. 1. B F 8.5 17 8 D A 20 10 9 G 4 H 6 8 E J C 2. J N 6 L 9.1 M 7.2 P 11.2 8 S Chapter 8 K Q 5 7 R 9.6 6 152 Geometry Practice Workbook with Examples Copyright © McDougal Littell Inc. All rights reserved. LESSON 8.5 CONTINUED EXAMPLE 2 NAME _________________________________________________________ DATE ____________ Practice with Examples For use with pages 488–496 Using the SAS Similarity Theorem Use the given lengths to prove that ABC ~ DEC. E 16 B 4C A SOLUTION 3 12 D Begin by finding the ratios of the lengths of the corresponding sides. BC 4 1 EC 16 4 AC 3 1 DC 12 4 So, the side lengths AC and BC of ABC are proportional to the corresponding side lengths DC and EC of DEC. The included angle in ABC is BCA; the included angle in DEC is ECD. Because these two angles are vertical angles, they are congruent. So, by the SAS Similarity Theorem, ABC ~ DEC. Exercises for Example 2 Prove that the two triangles are similar. 3. 4. E P D 5. 10 T Q 4 11 16 12.8 A 4 B U 6 N S 9 9 R 10 C R 6 V Chapter 8 20 M Copyright © McDougal Littell Inc. All rights reserved. Geometry Practice Workbook with Examples 153