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The Isosceles Triangle Theorems. 4-4. Isosceles Triangle:. Vertex angle. Leg. Leg. Base Angles. Base. The base of an Isosceles Triangle does NOT have to be at the “bottom”.
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Isosceles Triangle: Vertex angle Leg Leg Base Angles Base
The base of an Isosceles Triangle does NOT have to be at the “bottom”
Theorem 4-1 The Isosceles Triangle Theorem:If two sides of a triangle are congruent, then the angles opposite those sides are congruent. A C B
Proof of Isosceles Theorem A 1) Given 2) Def ∠ Bisector 3) Reflexive Prop 4) SAS 5) CPCTC C B D
Corollaries Corollary 1 An equilateral triangle is also equiangular. Corollary 2 An equilateral triangle has three 60° angles. Corollary 3 The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.
Example: 1 C 2 If m∠1 = 140, m∠2 = _____, m∠3 _____, m∠4 = _____ If m∠4 = 65, m∠3 = ______, m∠2 = _____, m∠1 = _______ 3 4 B A
B Example Proof 1 2 j k C A 1) Given 2) Def of isos. ∆ 3) Isos. ∆ Thm 4) If lines are ||, corr. ∠‘s ≅ 5) Substitution Prop (3,4)
Theorem 4-2 (CONVERSE)If two angles of a triangle are congruent, then the sides opposite those angles are congruent. CorollaryAn equiangular triangle is also equilateral.
