Holt CA Course Similar Figures and Proportions Preparation for NS1.3 Use proportions to solve problems (e.g., determine the value of N if =, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. California Standards

Holt CA Course Similar Figures and Proportions Corresponding angles of two or more polygons are in the same relative position. Corresponding sides of two or more figures are between corresponding angles. Similar figures have the same shape but not necessarily the same size. The symbol ~ means “is similar to.”

Holt CA Course 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.

Holt CA Course Similar Figures and Proportions Tell whether the triangles are similar. Teacher Example 1: Determining Whether Two Triangles Are Similar AC B 10 in 4 in 7 in AB corresponds to DE. BC corresponds to EF. AB DE = ? BC EF = ? AC DF Since the measures of the corresponding angles and the ratios of the corresponding sides are equivalent, the triangles are similar. Write ratios using the corresponding sides. Substitute the length of the sides. Simplify each ratio. = ? = ? AC corresponds to DF. = ? = ? The corresponding angles of the figures have equal measures. D E F 16 in 28 in 40 in

Holt CA Course Similar Figures and Proportions When naming similar figures, list the letters of the corresponding angles in the same order. Writing Math

Holt CA Course Similar Figures and Proportions Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar. Student Practice 1: AC B 9 in 3 in 7 in AB corresponds to DE. BC corresponds to EF. AB DE = ? BC EF = ? AC DF Since the measures of the corresponding angles and the ratios of the corresponding sides are equivalent, the triangles are similar. Write ratios using the corresponding sides. Substitute the length of the sides. Simplify each ratio. = ? = ? AC corresponds to DF. = ? = ? The corresponding angles of the figures have equal measures. D E F 9 in 21 in 27 in

Holt CA Course Similar Figures and Proportions For triangles, if the corresponding side lengths are all proportional, then the corresponding angles must have equal measures. For figures that have four or more sides, if the corresponding side lengths are all proportional, then the corresponding angles may or may not have equal measures.

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

Holt CA Course Similar Figures and Proportions Teacher Example 2 Continued: MN QR MN corresponds to QR. NO RS OP ST MP QT NO corresponds to RS. OP corresponds to ST. MP corresponds to QT.

Holt CA Course Similar Figures and Proportions Determine whether the ratios of the lengths of the corresponding sides are proportional. Teacher Example 2 Continued: Write ratios using corresponding sides. Substitute the length of the sides. Simplify each ratio. Since the ratios of the corresponding sides are equivalent, the figures are similar. MN QR = ? NO RS = ? OP ST = ? MP QT 6 9 = ? 8 12 = ? 4 6 = ? = 2323 = 2323 = 2323

Holt CA Course Similar Figures and Proportions A side of a figure can be named by its endpoints, with a bar above. AB Without the bar, the letters indicate the length of the side. Reading Math

Holt CA Course Similar Figures and Proportions Tell whether the figures are similar. Student Practice 2: The corresponding angles of the figures have equal measure. 100 m 80 m 60 m 47.5 m 80° 90° 125° 65° MP N O 400 m 320 m 190 m240 m 80°65° 90° 125° QT R S Write each set of corresponding sides as a ratio.

Holt CA Course Similar Figures and Proportions Student Practice 2: Continued MN QR MN corresponds to QR. NO RS OP ST MP QT NO corresponds to RS. OP corresponds to ST. MP corresponds to QT. 100 m 80 m 60 m 47.5 m 80° 90° 125° 65° MP N O 400 m 320 m 190 m240 m 80°65° 90° 125° QT R S

Holt CA Course Similar Figures and Proportions Determine whether the ratios of the lengths of the corresponding sides are proportional. Student Practice 2: Continued Write ratios using corresponding sides. Substitute the length of the sides. Simplify each ratio. MN QR = ? NO RS = ? OP ST = ? MP QT = ? = ? = ? = 1414 = 1414 = 1414 Since the measure of the corresponding angles are equal and the ratios of the corresponding sides are equivalent, the figures are similar. 100 m 80 m 60 m 47.5 m 80° 90° 125° 65° MP N O 400 m 320 m 190 m240 m 80°65° 90° 125° QT R S