4 4 Similar Figures and Proportions Warm Up

4 -4 Similar Figures and Proportions Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

4 -4 Similar Figures and Proportions Warm Up Find the cross products, and then tell whether the ratios are equal. 1. 16, 40 6 15 2. 3. 3, 8 8, 9 18 46 24 27 4. 28 , 42 12 18 240 = 240; equal 138 = 144; not equal 216 = 216; equal 504 = 504; equal

4 -4 Similar Figures and Proportions Problem of the Day Every 8 th telephone pole along a road has a red band painted on it. Every 14 th pole has an emergency call phone on it. What is the number of the first pole with both a red band a call phone? 56

4 -4 Similar Figures and Proportions Learn to use ratios to determine if two figures are similar.

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

4 -4 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 -4 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 -4 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 -4 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 -4 Similar Figures and Proportions Additional Example 1: Determining Whether Two Triangles Are Similar Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar. E AB corresponds to DE. 16 in A 10 in C 28 in BC corresponds to EF. D 4 in 7 in 40 in F AC corresponds to DF. B 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 -4 Similar Figures and Proportions Check It Out: Example 1 Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar. E AB corresponds to DE. 9 in C A 21 in BC corresponds to EF. D 3 in 7 in 27 in F AC corresponds to DF. B AB ? BC ? AC = = Write ratios using the corresponding sides. DE EF DF 3 ? 7 ? 9 Substitute the length of the sides. = = 9 27 21 ? 1 1 = Simplify each ratio. = 3 3 3 Since the ratios of the corresponding sides are equivalent, the triangles are similar.

4 -4 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 -4 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 -4 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 -4 Similar Figures and Proportions Check It Out: Example 2 Tell whether the figures are similar. M P 100 m 80° 60 m 90° N Q 65° 125° 80 m 47. 5 m O T 400 m 80° 65° 240 m R 90° 125° 320 m The corresponding angles of the figures have equal measure. 190 m S Write each set of corresponding sides as a ratio.

4 -4 Similar Figures and Proportions Check It Out: Example 2 Continued M P 100 m 80° 60 m 90° N Q 65° 125° 80 m 47. 5 m O T 65° 240 m R 90° 125° 320 m MN corresponds to QR. NO NO corresponds to RS. RS 400 m 80° MN QR 190 m S OP ST OP corresponds to ST. MP QT MP corresponds to QT.

4 -4 Similar Figures and Proportions Check It Out: Example 2 Continued Determine whether the ratios of the lengths of the corresponding sides are proportional. M P 100 m 80° 60 m 90° N Q 65° 125° 80 m 47. 5 m O T 400 m 80° 65° 240 m R 90° 125° 320 m 190 m ? corresponding ? MP ? NO OP = MNratios Write using = = sides. ST QR RS QT 60 ? 80 100 ? 47. 5 of Substitute the length ? the 240 = 320 = 190 = 400 sides. ? 1? 1 1= Simplify= each=ratio. 4 4 S Since the ratios of the corresponding sides are equivalent, the figures are similar.

4 -4 Similar Figures and Proportions Lesson Quizzes Standard Lesson Quiz for Student Response Systems

4 -4 Similar Figures and Proportions Lesson Quiz: Part I Tell whether the figures are similar. 1. 59° 35° similar 59° 86° 35°

4 -4 Similar Figures and Proportions Lesson Quiz: Part II Tell whether the figures are similar. 2. 119° 55° 107° 80° not similar 107° 79° 135° 38°

4 -4 Similar Figures and Proportions Lesson Quiz for Student Response Systems 1. Which of the following triangles are similar? A. C. B. D.

4 -4 Similar Figures and Proportions Lesson Quiz for Student Response Systems 2. Which of the following figures are similar? A. B.