Lesson 6.5, For use with pages 388-395 Determine whether the two triangles are similar. 1. ABC: m A = 90º, m B = 44º; DEF : m D = 90º, m E = 46º. ANSWER similar 2. ABC: m A = 132º, m B = 24º; DEF : m D = 90º, m F = 24º. ANSWER not similar

Lesson 6.5, For use with pages 388-395 3. Solve = . 12 6 x – 1 8 ANSWER 5

EXAMPLE 1 Use the SSS Similarity Theorem Is either DEF or GHJ similar to ABC? SOLUTION Compare ABC and DEF by finding ratios of corresponding side lengths. Shortest sides AB DE 4 3 8 6 =

EXAMPLE 1 Use the SSS Similarity Theorem Longest sides CA FD 4 3 16 12 = Remaining sides BC EF 4 3 12 9 = All of the ratios are equal, so ABC ~ DEF. ANSWER Compare ABC and GHJ by finding ratios of corresponding side lengths. Shortest sides AB GH 8 = 1

EXAMPLE 1 Use the SSS Similarity Theorem Longest sides CA JG 16 = 1 Remaining sides BC HJ 6 5 12 10 = The ratios are not all equal, so ABC and GHJ are not similar. ANSWER

Use the SSS Similarity Theorem EXAMPLE 2 Use the SSS Similarity Theorem ALGEBRA Find the value of x that makes ABC ~ DEF. SOLUTION STEP 1 Find the value of x that makes corresponding side lengths proportional. 4 12 = x –1 18 Write proportion.

Use the SSS Similarity Theorem EXAMPLE 2 Use the SSS Similarity Theorem 4 18 = 12(x – 1) Cross Products Property 72 = 12x – 12 Simplify. 7 = x Solve for x. Check that the side lengths are proportional when x = 7. STEP 2 BC = x – 1 = 6 AB DE BC EF = ? 6 18 4 12 =

Use the SSS Similarity Theorem EXAMPLE 2 Use the SSS Similarity Theorem DF = 3(x + 1) = 24 AB DE AC DF = ? 8 24 4 12 = When x = 7, the triangles are similar by the SSS Similarity Theorem. ANSWER

GUIDED PRACTICE for Examples 1 and 2 1. Which of the three triangles are similar? Write a similarity statement. MLN ~ ZYX. ANSWER

GUIDED PRACTICE for Examples 1 and 2 2. The shortest side of a triangle similar to RST is 12 units long. Find the other side lengths of the triangle. ANSWER 15, 16.5

EXAMPLE 3 Use the SAS Similarity Theorem 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?

Use the SAS Similarity Theorem EXAMPLE 3 Use the SAS Similarity Theorem SOLUTION 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 = The lengths of the sides that include A and F are proportional.

EXAMPLE 3 Use the SAS Similarity Theorem ANSWER So, by the SAS Similarity Theorem, ABC ~ FGH. Yes, you can make the right end similar to the left end of the shelter.

Tell what method you would use to show that the triangles are similar. EXAMPLE 4 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 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.

GUIDED PRACTICE for Examples 3 and 4 3. SRT ~ PNQ Explain how to show that the indicated triangles are similar. ANSWER R  N and = = , therefore the triangles are similar by the SAS Similarity Theorem. SR PN RT NQ 4 3

GUIDED PRACTICE for Examples 3 and 4 4. XZW ~ YZX Explain how to show that the indicated triangles are similar. XZ YZ WZ 4 3 = WX XY WZX  XZY and therefore the triangles are similar by either SSS or SAS Similarity Theorems. ANSWER