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6.5 Prove Triangles Similar by SSS and SAS Goal Use the SSS and SAS Similarity Theorems. Your Notes THEOREM 6.2: SIDE-SIDE-SIDE (SSS) SIMILARITY THEOREM If the corresponding side lengths of two triangles are _proportional_, then the triangles are similar. BC CA If AB = = then ABC ~ RST. ST TR RS 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 = 8 = _2_ DE 4 Longest sides CA = 12 = 3 8 2 FD Remaining sides BC = 9 = 3 A 6 2 EF When using the SSS Similarity Theorem, compare the shortest sides, the longest sides, and then the remaining sides. All the ratios are _not equal_ , so ABC and DEF are _not similar_. Compare ABC and GHJ by finding ratios of corresponding side lengths. Shortest sides 1 AB 8 = = GH 16 2 Longest sides CA 12 1 = = JG 24 2 D Remaining sides BC = 9 = 1 18 2 A HJ All the ratios are _equal_ , so _ABC_ ~ _GHJ_. Your Notes Example 2 Use the SSS Similarity Theorem Find the value of x that makes ABC ~ DEF. Solution Step 1 Find the value of x that makes corresponding side lengths proportional. 4 10 =x 3 20 Write proportion. 4. _20_ = _10_(x 3) Cross Products Property _80_ = _10_x _30_ Simplify. _11_ = x Solve for x. Check that the side lengths are proportional when x = _11_. DF = 2x + 3 = _25_ AB = x 3 = _8_ Step 2 BC =? AB DE EF 8 4 = 20 10 10 BC ? AC 4 = = EF DF 10 25 When x = _11_, the triangles are similar by the _SSS Similarity Theorem_. Checkpoint Complete the following exercises. 1. Which of the three triangles are similar? PQR ZXY 2. Suppose AB is not given in ABC. What length for AB would make ABC similar to QRP? AB = 36 Your Notes THEOREM 6.3: 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. If X M , and ZX XY = , then XYZ MNP. PM MN Example 3 Use the SAS Similarity Theorem Birdfeeder You are drawing a design for a birdfeeder. Can you construct the top so it is similar to the bottom using the angle measure and lengths shown? Solution Both m_B_ and m_E_ equal 87°, so _B_ _E_. Next, compare the ratios of the lengths of the sides that include B and E. BC 32 8 AB EF 20 5 DE The lengths of the sides that include B and E are _proportional_. So, by the _SAS Similarity Theorem_, ABC ~ DEF. Yes, you can make the top similar to the bottom. Checkpoint Complete the following exercise. 3. In Example 3, suppose you use equilateral triangles on the top and bottom. Are the top and bottom similar? Explain. Yes, the top and bottom are similar. If the side length of the top is a and the side a length of the bottom is b, the ratios of the side lengths are and the angles are all b 60°. The triangles are similar by SAS or SSS. Your Notes TRIANGLE SIMILARITY POSTULATE AND THEOREMS AA Similarity Postulate If A D and B E, then ABC ~ DEF. SSS Similarity Theorem If AB = BC = AC , then ABC ~ DEF. DE EF DF AB AC SAS Similarity Theorem If A D and = , then ABC DEF. DE DF 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. To identify corresponding parts, redraw the triangles so that the corresponding parts have the same orientation. Shorter sides QR 9 3 RS 12 4 Longer sides PR 18 3 RT 24 4 The corresponding side lengths are _proportional_. The included angles PRQ and TRS are _congruent_ because they are _vertical_ angles. So, PQR ~ TSR by the _SAS Similarity Theorem_. Checkpoint Complete the following exercise. 4. Explain how to show JKL ~ LKM. Show that the corresponding side lengths are proportional, then use the SSS Similarity Theorem to show JKL ~ LKM. Homework ________________________________________________________________________ ________________________________________________________________________