4 8 Isosceles and Equilateral Triangles Objectives Prove

4 -8 Isosceles and Equilateral Triangles Objectives Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Recall that an isosceles triangle has at least two congruent sides. The congruent sides are called the legs. The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base, and the base angles are the two angles that have the base as a side. 3 is the vertex angle. 1 and 2 are the base angles. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Holt Geometry

4 -8 Isosceles and Equilateral Triangles Reading Math The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent. ” Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 1: Astronomy Application The length of YX is 20 feet. Explain why the length of YZ is the same. The m YZX = 180 – 140, so m YZX = 40°. Since YZX X, ∆XYZ is isosceles by the Converse of the Isosceles Triangle Theorem. Thus YZ = YX = 20 ft. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 2 A: Finding the Measure of an Angle Find m F = m D = x° Isosc. ∆ Thm. m F + m D + m A = 180 ∆ Sum Thm. Substitute the x + 22 = 180 given values. Simplify and subtract 2 x = 158 22 from both sides. x = 79 Divide both sides by 2. Thus m F = 79° Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 2 B: Finding the Measure of an Angle Find m G. m J = m G Isosc. ∆ Thm. (x + 44) = 3 x 44 = 2 x Substitute the given values. Simplify x from both sides. Divide both sides by 2. Thus m G = 22° + 44° = 66°. x = 22 Holt Geometry

4 -8 Isosceles and Equilateral Triangles The following corollary and its converse show the connection between equilateral triangles and equiangular triangles. Holt Geometry

4 -8 Isosceles and Equilateral Triangles Holt Geometry

4 -8 Isosceles and Equilateral Triangles Example 3 A: Using Properties of Equilateral Triangles Find the value of x. ∆LKM is equilateral. Equilateral ∆ equiangular ∆ (2 x + 32) = 60 2 x = 28 x = 14 Holt Geometry The measure of each of an equiangular ∆ is 60°. Subtract 32 both sides. Divide both sides by 2.

4 -8 Isosceles and Equilateral Triangles Example 3 B: Using Properties of Equilateral Triangles Find the value of y. ∆NPO is equiangular. Equiangular ∆ equilateral ∆ 5 y – 6 = 4 y + 12 y = 18 Holt Geometry Definition of equilateral ∆. Subtract 4 y and add 6 to both sides.