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7.4

7.4. OBJ: Show that two triangles are similar using the SSS and SAS Similarity Theorems. Side-Side-Side (SSS) Similarity Theorem. If the corresponding sides of two triangles are proportional, then the triangles are similar. Example 1. Use the SSS Similarity Theorem.

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7.4

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  1. 7.4 OBJ: Show that two triangles are similar using the SSS and SAS Similarity Theorems.

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

  3. Example 1 Use the SSS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of Triangle B to Triangle A. SU 6 6 ÷ 6 1 = = = PR 12 12 ÷ 6 2 UT 5 5 ÷ 5 1 = = = RQ 10 ÷ 5 2 10 SOLUTION Find the ratios of the corresponding sides. All three ratios are equal. By the SSS Similarity Theorem, PQR ~ STU. TS 4 4 ÷ 4 1 = = = QP 8 ÷ 4 2 8 The scale factor of Triangle B to Triangle A is 1/2.

  4. Example 2 Use the SSS Similarity Theorem Is eitherDEF orGHJ similar to ABC? SOLUTION 1. Look at the ratios of corresponding sides in ABCandDEF. Shortest sides Longest sides Remaining sides EF DE FD 4 6 2 2 2 8 = = = = = = CA AB BC 3 3 3 9 6 12 Because all of the ratios are equal, ABC ~ DEF. BUT: ANSWER

  5. Example 2 Use the SSS Similarity Theorem Because the ratios are not equal, ABC andGHJ are not similar. ANSWER 2. Look at the ratios of corresponding sides in ABCandGHJ. Shortest sides Longest sides Remaining sides HJ GH JG 10 6 1 7 14 = = = = = CA AB BC 9 6 1 6 12

  6. Checkpoint Use the SSS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. 1. 2.

  7. Checkpoint Use the SSS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. 1. ANSWER yes;ABC ~ DFE 2. no ANSWER

  8. 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 that include these angles are proportional, then the triangles are similar.

  9. Example 3 Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. SOLUTION FE 10 5 DF 5 Shorter sides Longer sides = CB 6 = 3 = AC 3 The lengths of the sides that include CandFare proportional. By the SAS Similarity Theorem, ABC ~ DEF. ANSWER

  10. Example 4 Similarity in Overlapping Triangles Show thatVYZ ~ VWX. SOLUTION V  V by the Reflexive Property of Congruence. Shorter sides Longer sides XV VW 4 5 1 1 5 4 = = = = = = ZV VY 5 + 10 4 + 8 3 3 12 15

  11. Example 4 Continuation…. The lengths of the sides that include Vare proportional. By the SAS Similarity Theorem, VYZ ~VWX. ANSWER

  12. Checkpoint . Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 12 8 No; H M but 6 8 ANSWER ≠

  13. Checkpoint ANSWER , Yes; P P,and the so PQR ~PST by the SAS Similarity Theorem. Use the SAS Similarity Theorem Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 4. 3 5 1 1 PR PQ ; = = = = 2 2 6 10 PT PS

  14. JKL~PNM ABC is not similar to DEF. ANSWER ANSWER Review: Determine whether the triangles are similar. If they are similar, write a similarity statement. 1. 2.

  15. x = 15 ANSWER 3. Find the value ofx.

  16. Homework • Worksheet 7.4A

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