6.5 Prove Triangles Similar by SSS and SAS Hubarth Geometry

Side-side-side Similarity Theorem (SSS) ~ A F G H C B

Ex 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. P S B 6 4 B 8 12 A T U 5 R Q 10 Solution

Ex 2. Use the SSS Similarity Theorem 12 14 A C E G J 6 4 6 6 9 10 D F 8 H B Solution

Ex 3 Use the SSS Similarity Theorem Find the value of x that makes corresponding side lengths proportional. DF = 3(x + 1) = 24 4 12 = x –1 18 AB DE AC DF = ? 8 24 4 12 = 4 18 = 12(x – 1) 72 = 12x – 12 7 = x

Side-Angle-Side Similarity Theorem (SAS) X M P N Z Y

Shorter sides Longer sides Ex 4 Use the SAS Similarity Theorem 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? Both m A and m F equal = 53°, so A F. Next, compare the ratios of the lengths of the sides that include A and F. Shorter sides Longer sides AB FG 3 2 9 6 = AC FH 3 2 15 10 = So, by the SAS Similarity Theorem, ABC ~ FGH. Yes, you can make the right end similar to the left end of the shelter.

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

Practice 1. Determine whether the triangles are similar. If they are similar, write a similarity statement. J A 15 10 R D A. B F B. 12 12 13 8 18 15 12 12 E K L C 5 S T 9 2. Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. P 5 3 Q R 3 5 S T