4.2.5 Isosceles and Equilateral Triangles Key Objectives

• Prove theorems about isosceles and equilateral triangles. • Apply properties of isosceles and equilateral triangles. Key Terms • The congruent sides of an isosceles triangle are called the legs . • The vertex angle is the angle formed by the legs (congruent sides) of an isosceles triangle. • The side opposite of the vertex angle in an isosceles triangle is called the base . • The base angles of an isosceles triangle are the two angles that have the base as a side. Theorems, Postulates, Corollaries, and Properties • Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent. • Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

• Equilateral Triangle Theorem If a triangle is equilateral, then it is equiangular. • Equiangular Triangle Theorem If a triangle is equiangular, then it is equilateral. Example 1 Applying the Isosceles Triangle Theorem

An isosceles triangle is a triangle with at least two congruent sides. The Isosceles Triangle Theorem states that when a triangle has two congruent sides, it can be assumed that the angles opposite of those sides are also congruent. The Converse of Isosceles Triangle Theorem states the reverse of the Isosceles Triangle Theorem. By the converse, when a triangle has two congruent angles, it can be assumed that the sides opposite of those angles are also congruent. The figure is given for this proof, along with m ∠ ABE , m ∠ CDE , and the fact that C is the midpoint between B and D . However, notice that this proof does not use the fact that C is a midpoint. Make a plan for the proof. The conclusion to be proven is that BE = ED . Notice that BE and ED are the measures of two sides of △ BED . So, show that ∠ CBE is congruent to ∠ CDE . From there, the Converse of Isosceles Triangle Theorem can be applied because △ BED contains two congruent angles, ∠ CBE and ∠ CDE . So, by the Converse of Isosceles Triangle Theorem, the sides of △ BED that are opposite of ∠ CBE and ∠ CDE must also be congruent.

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