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Transcript
Theorem 8-5: Triangle-Angle-Bisector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. Chapter 8: Similarity Postulate 8-1: Angle-Angle Similarity (AA ~) Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Theorem 8-1: Side-Angle-Side Similarity (SAS ~) Thm Theorem 8-6: Perimeters and Areas of Similar Figures a If the similarity ratio of two similar figures is , then b a (1) the ratio of their perimeters is and b a2 (2) the ratio of their areas is b2 If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar. . Theorem 8-2: Side-Side-Side Similarity (SSS ~) Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. Theorem 8-3 Theorem 8-4: Side-Splitter Theorem The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other. If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Corollary 1 Corollary The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse. If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. Converse Corollary 2 The altitude to the hypotenuse of a right triangle separates the hypotenuse in such a way that the length of each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse. If a line divides two sides of a triangle proportionally, then it is parallel to the third side.