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4.5: Isosceles and Equilateral Triangles does not not do do to to dwell dwell on on dreams… dreams… and and forget forget to to live. live. ItIt does -Dumbledore Isosceles Triangles The congruent sides are the LEGS. The third side is the BASE. The two congruent sides form the VERTEX ANGLE. The other two angles are the BASE ANGLES. Theorems Theorem 4.3: Isosceles Triangle Theorem – If two sides of a triangle are congruent, then the angles opposite those sides are congruent. A B Proof of the Isosceles Triangle Theorem Given : XY XZ , XB bisects YXZ Prove : Y Z Statements 1. XY XZ , Reasons 1. XB bisects YXZ 1 2 2. 3. XB XB 3. 4. XYB XZB 4. 5. Y Z 5. 2. Theorems Theorem 4.4: CONVERSE of Isosceles Triangle Theorem – If two angles of a triangle are congruent, then the sides opposite the angles are congruent. AC BC Theorems Theorem 4.5: The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. CD AB and CD bisects AB Applications Explain why each statement is true. a. WVS S b. TR TS Applications Find the value of y. Corollaries corollary – a statement that follows immediately from a theorem Corollary to Theorem 4.3: If a triangle is equilateral, then the triangle is equiangular. X Y Z Corollary to Theorem 4.4: If a triangle is equiangular, then the triangle is equilateral. XY YZ ZX Applications Find the value of the variables. 1) 2) 4.5: Isosceles and Equilateral Triangles Homework 4.5: #1-13, 15, 20, 22, 28, 30 I don't know half of you half as well as I should like, and I like less I don't know half of you half as well as I should like, and I like less than half of you half as well as you deserve. than half of you half as well as you deserve. -Bilbo