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Isosceles and Equilateral Triangles Geometry (Holt 4-9) K.Santos Parts of an Isosceles Triangle Isosceles triangle—is a triangle with at least two congruent sides A B C Legs—are the congruent sides and Vertex angle—angle formed by the legs <A Base—side opposite the vertex angle Base angles—two angles that have the base as a side < B and < C Isosceles Triangle Theorem (4-9-1) If two sides of a triangle are congruent, then the angles opposite the sides are congruent. A B C Given: ≅ Then: <A ≅ <C congruent sides congruent angles Converse of the Isosceles Triangle Theorem (4-9-2) If two angles of a triangle are congruent, then the sides opposite those angles are congruent. A B C Given: <A ≅ <C Then: ≅ congruent angles congruent sides Example—finding angle measures A Find the measure of <C. C m< B = m< A = x m < C + m < B + m < A = 180 x + 50 + 50 = 180 x + 100 = 180 x = 80 which means m< C = 80° 50° B Example—finding angle measures A Find the measure of < C. 50° x C m< C = m< B = x m < C + m < B + m < A = 180 x + x + 50 = 180 2x + 50 = 180 2x = 130 x = 65 which means m < C = 65° B Example—finding angle measures (algebraic) S Find x. x + 38° T 3x° R m<R=m<S 3x = x + 38 2x = 38 x = 19 Corollary(4-9-3)— Equilateral Triangle If a triangle is equilateral, then it is equiangular. M N Given: ≅ ≅ Then: <M ≅ <N ≅ <O equilateral O 180/3 = 60° equiangular Corollary (4-9-4)— Equiangular Triangle If a triangle is equiangular, then it is equilateral. M N Given: <M ≅ <N ≅ <O O Then: ≅ ≅ equiangular equilateral Example—Finding angles Find x. G 4x+12 H I triangle is equilateral-----equiangular each angle is 60° 4x + 12 = 60 4x = 48 x = 12° Example—finding sides J Find t. 3t + 3 K 5t – 9 L Triangle is equiangular---equilateral (all equal sides) ≅ 5t – 9 = 3t + 3 2t – 9 = 3 2t = 12 t=6 Example—Multiple Triangles Find the measures of the numbered angles. m< 3 = m< 4 x + x + 80 = 180 2x + 80 = 180 2x = 100 x = 50 so m <3 and m < 4 = 50 m< 1 and m< 4 are supplementary m< 1 + m < 4 = 180 m< 1 + 50 = 180 m< 1 = 130 m< 1 + m< 2 + m< 5 = 180 with m< 2 = m< 5 130 + y + y = 180 130 + 2y = 180 2y = 50 y = 25 so m< 2 = 25, and m< 5 = 25 80 5 3 4 1 2