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7 THE NATURE OF GEOMETRY Copyright © Cengage Learning. All rights reserved. 7.4 Similar Triangles Copyright © Cengage Learning. All rights reserved. Similar Triangles Congruent figures have exactly the same size and shape. However, it is possible for figures to have exactly the same shape without necessarily having the same size. Such figures are called similar figures. If ABC is similar to DEF, we write ABC ~ DEF 3 Similar Triangles Similar triangles are shown in Figure 7.38. Similar Triangles Figure 7.38 m A = m D, so these are corresponding angles. m B = m E, so these are corresponding angles. m C = m F, so these are corresponding angles. 4 Similar Triangles Side BC is opposite A and side EF is opposite so we say that BC corresponds to EF. D, AC corresponds to DF. AB corresponds to DE. The corresponding angles of similar triangles are those angles that have equal measure. The corresponding sides are those sides that are opposite equal angles. 5 Similar Triangles Even though corresponding angles are equal, corresponding sides do not need to have the same length. If they do have the same length, the triangles are congruent. However, when they are not the same length, we can say they are proportional. From Figure 7.38 we see that the lengths of the sides are labeled a, b, c and d, e, f. Similar Triangles Figure 7.38 6 Similar Triangles When we say the sides are proportional, we mean Primary ratios: Reciprocals: 7 Similar Triangles We summarize with an important property of similar triangles called the similar triangle theorem. 8 Example 2 – Find lengths given similar triangles Given the similar triangles in Figure 7.40, find the unknown lengths marked b and c. Given ABC ~ ABC Figure 7.40 9 Example 2 – Solution Since corresponding sides are proportional (other proportions are possible), we have 10 Similar Triangles There is a relationship between the sizes of the angles of a right triangle and the ratios of the lengths of the sides. In a right triangle, the side opposite the right angle is called the hypotenuse. Each of the acute angles of a right triangle has one side that is the hypotenuse; the other side of that angle is called the adjacent side. (See Figure 7.42.) A right triangle Figure 7.42 11 Similar Triangles In ABC with right angle at C: The hypotenuse is c; The side adjacent to A is b; The side adjacent to B is a. The side opposite A is a, and the side opposite B is b. 12