4 8 Similar Figures and Proportions Vocabulary similar

4 -8 Similar Figures and Proportions Vocabulary similar corresponding sides corresponding angles

4 -8 Similar Figures and Proportions Similar figures are figures that have the same shape but not necessarily the same size. The symbol ~ means “is similar to. ”

4 -8 Similar Figures and Proportions Corresponding angles of two or more similar polygons are in the same relative position. Corresponding sides of two or more similar polygons are in the same relative position. When naming similar figures, list the corresponding angles in the same order. For the triangles below, ∆ABC ~ ∆DEF. E B A C D F

4 -8 Similar Figures and Proportions SIMILAR FIGURES Two figures are similar if • the measures of their corresponding angles are equal. • the ratios of the lengths of the corresponding sides are proportional.

4 -8 Similar Figures and Proportions Reading Math A side of a figure can be named by its endpoints, with a bar above such as; AB Without the bar, the letters indicate the length of the side.

4 -8 Similar Figures and Proportions Additional Example 1: Determining Whether Two Triangles Are Similar Tell whether the triangles are similar. AB corresponds to DE. BC corresponds to EF. AC corresponds to DF. AB ? BC ? AC = = Write ratios using the corresponding sides. DE EF DF 4 ? 7 ? 10 Substitute the length of the sides. = = 16 40 28 ? 1 1 = Simplify each ratio. = 4 4 4 Since the ratios of the corresponding sides are equivalent, the triangles are similar.

4 -8 Similar Figures and Proportions Check It Out: Example 1 Tell whether the triangles are similar. A 3 in. B 9 in. C 7 in. 9 in. D E 21 in. 27 in. F AB corresponds to DE; BC corresponds to EF; AC corresponds to DF. AB ? BC ? AC = = DE EF DF ; 3 = 7 = 9 ; the triangles are similar. 9 21 27

4 -8 Similar Figures and Proportions Additional Example 2: Determining Whether Two Four-Sided Figures are Similar Tell whether the figures are similar. The corresponding angles of the figures have equal measure. Write each set of corresponding sides as a ratio.

4 -8 Similar Figures and Proportions Additional Example 2 Continued MN QR MN corresponds to QR. NO NO corresponds to RS. RS OP ST OP corresponds to ST. MP QT MP corresponds to QT.

4 -8 Similar Figures and Proportions Additional Example 2 Continued Determine whether the ratios of the lengths of the corresponding sides are proportional. ? OP ? MP ? NO = MN = = ST QR RS QT Write ratios using corresponding sides. 6? 8 ? = = 9 12 Substitute the length of the sides. 4 ? 10 6 = 15 ? 2= 2= ? 2 3 3 Simplify each ratio. Since the ratios of the corresponding sides are equivalent, the figures are similar.

4 -8 Similar Figures and Proportions Check It Out: Example 2 A Tell whether the figures are similar. B A 10 cm 75° 105° 14 cm L 7 cm 75° 105° D 10 cm C 5 cm 115° O 65° 115° 5 cm M 7 cm N No; the corresponding angles of the figures do not have equal measures.

4 -8 Similar Figures and Proportions Check It Out: Example 2 B Tell whether the figures are similar. P 8 ft 70° Q 70° 10 ft S 110°R 5 ft T 6 ft U 70° 7. 5 ft 110° W V 110° 3. 75 ft Yes, the corresponding angles of the figures have equal measures, and PQ SR 10 PS QR , 8 5 10 = = = TU WV TW UV 6 3. 75 7. 5 so the sides are proportional.

4 -8 Similar Figures and Proportions Lesson Quiz Tell whether the triangles are similar. 1. similar Tell whether the figures are similar. 2. not similar