1. Bell Ringer 6. 7.

2. Similar Polygons Two polygons are similar polygons if corresponding angles are congruent and corresponding side length are proportional.

3. Example 1 Use Similarity Statements  PRQ ~  STU. List all pairs of congruent angles.a. Write the ratios of the corresponding sides in a statement of proportionality. b. Check that the ratios of corresponding sides are equal. c. SOLUTION  P   S,  R   T, and  Q   U. a. PR ST RQ TU QP US = = b. c. PR ST 10 8 5 4 = =, RQ TU 20 16 5 4 = =, and QP US 15 12 5 4 = =. The ratios of corresponding sides are all equal to 5 4.

4. If two polygons are similar, then the ratios of the lengths of the two corresponding sides is called the scale factor.

5. Example 2 Determine Whether Polygons are Similar Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of Figure B to Figure A. SOLUTION Check whether the corresponding angles are congruent. 1. From the diagram, you can see that  G   M,  H   K, and  J   L. Therefore, the corresponding angles are congruent.

6. Example 2 Determine Whether Polygons are Similar GH MK 9 12 9 ÷ 3 12 ÷ 3 = = 3 4 = HJ KL 12 16 12 ÷ 4 16 ÷ 4 = = 3 4 = JG LM 15 20 15 ÷ 5 20 ÷ 5 = = 3 4 = All three ratios are equal, so the corresponding side lengths are proportional. ANSWER By definition, the triangles are similar.  GHJ ~  MKL. The scale factor of Figure B to Figure A is 3 4. Check whether the corresponding side lengths are proportional.2.

7. Now You Try Determine Whether Polygons are Similar Determine whether the polygons are similar. If they are similar, write a similarity statement and find the scale factor of Figure B to Figure A. 1. 2. ANSWER yes;  XYZ ~  DEF ; 2 3 ANSWER no 6 9 10 12 ≠

8. Example 3 Use Similar Polygons  RST ~  GHJ. Find the value of x. SOLUTION Because the triangles are similar, the corresponding side lengths are proportional. To find the value of x, you can use the following proportion. RS GH TR JG = Write proportion. 10 15 x 9 = Substitute 15 for GH, 10 for RS, 9 for JG, and x for TR. Cross product property 15 · x = 10 · 9

9. Example 3 Use Similar Polygons 15 15x 15 90 = Divide each side by 15. 15x = 90 Multiply. Simplify. x = 6

10. Example 4 Perimeters of Similar Polygons The outlines of a pool and the patio around the pool are similar rectangles. Find the ratio of the length of the patio to the length of the pool. a. Find the ratio of the perimeter of the patio to the perimeter of the pool. b. The ratio of the length of the patio to the length of the pool isa. SOLUTION length of pool length of the patio 32 feet 48 feet = 32 ÷ 16 48 ÷ 16 ==. 2 3

11. Example 4 Perimeters of Similar Polygons perimeter of pool perimeter of patio 96 feet 144 feet = 96 ÷ 48 144 ÷ 48 ==. 2 3 b. The perimeter of the patio is 2(24) + 2(48) = 144 feet. The perimeter of the pool is 2(16) + 2(32) = 96 feet. The ratio of the perimeter of the patio to the perimeter of the pool is

12. Now You Try Use Similar Polygons ANSWER 18 ANSWER 2 1 Find the ratio of the perimeter of  STU to the perimeter of  PQR. 4. 3. Find the value of x. In the diagram,  PQR ~  STU.

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