OBJ: Show that two triangles are similar using the SSS and SAS 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.

Find the ratios of the corresponding sides. SOLUTION 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. Find the ratios of the corresponding sides. SOLUTION All three ratios are equal. PR SU 12 6 12 ÷ 6 6 ÷ 6 = 2 1 RQ UT 10 5 10 ÷ 5 5 ÷ 5 QP TS 8 4 8 ÷ 4 4 ÷ 4 By the SSS Similarity Theorem, PQR ~ STU. The scale factor of Triangle B to Triangle A is 1/2. 4

Is either DEF or GHJ similar to ABC? Example 2 Use the SSS Similarity Theorem Is either DEF or GHJ similar to ABC? SOLUTION Look at the ratios of corresponding sides in ABC and DEF. 1. Shortest sides = 6 4 AB DE 3 2 Longest sides 12 8 CA FD Remaining sides 9 BC EF ANSWER Because all of the ratios are equal, ABC ~ DEF. BUT: 5

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

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

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. ANSWER no

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.

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

V  V by the Reflexive Property of Congruence. Example 4 Similarity in Overlapping Triangles Show that VYZ ~ VWX. SOLUTION V  V by the Reflexive Property of Congruence. Shorter sides 4 + 8 4 = VY VW 12 3 1 Longer sides 5 + 10 5 ZV XV 15 11

The lengths of the sides that include V are proportional. Example 4 Continuation…. The lengths of the sides that include V are proportional. ANSWER By the SAS Similarity Theorem, VYZ ~ VWX. 12

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

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

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

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

Homework Worksheet 7.4A