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4-8 Triangles 4-8 Isosceles Isoscelesand and Equilateral Equilateral Triangles Lesson Presentation Holt Geometry Holt Geometry 4-8 Isosceles and Equilateral Triangles 1. Find each angle measure. 60°; 60°; 60° True or False. If false explain. 2. Every equilateral triangle is isosceles. True 3. Every isosceles triangle is equilateral. False; an isosceles triangle can have only two congruent sides. Holt Geometry 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 Vocabulary legs of an isosceles triangle vertex angle base base angles 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 mYZX = 180 – 140, so mYZX = 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 2A: Finding the Measure of an Angle Find mF. mF = mD = x° Isosc. ∆ Thm. mF + mD + mE = 180 ∆ Sum Thm. Substitute the x + x + 22 = 180 given values. Simplify and subtract 2x = 158 22 from both sides. x = 79 Divide both sides by 2. Thus mF = 79° Holt Geometry 4-8 Isosceles and Equilateral Triangles Example 2B: Finding the Measure of an Angle Find mG. mJ = mG Isosc. ∆ Thm. (x + 44) = 3x 44 = 2x Substitute the given values. Simplify x from both sides. Divide both sides by 2. Thus mG = 22° + 44° = 66°. x = 22 Holt Geometry 4-8 Isosceles and Equilateral Triangles Check It Out! Example 2A Find mH. mH = mG = x° Isosc. ∆ Thm. mH + mG + mF = 180 ∆ Sum Thm. Substitute the x + x + 48 = 180 given values. Simplify and subtract 2x = 132 48 from both sides. x = 66 Divide both sides by 2. Thus mH = 66° Holt Geometry 4-8 Isosceles and Equilateral Triangles Check It Out! Example 2B Find mN. mP = mN Isosc. ∆ Thm. (8y – 16) = 6y 2y = 16 y = 8 Substitute the given values. Subtract 6y and add 16 to both sides. Divide both sides by 2. Thus mN = 6(8) = 48°. 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 3A: Using Properties of Equilateral Triangles Find the value of x. ∆LKM is equilateral. Equilateral ∆ equiangular ∆ (2x + 32) = 60 2x = 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 3B: Using Properties of Equilateral Triangles Find the value of y. ∆NPO is equiangular. Equiangular ∆ equilateral ∆ 5y – 6 = 4y + 12 y = 18 Holt Geometry Definition of equilateral ∆. Subtract 4y and add 6 to both sides. 4-8 Isosceles and Equilateral Triangles Check It Out! Example 3 Find the value of JL. ∆JKL is equiangular. Equiangular ∆ equilateral ∆ 4t – 8 = 2t + 1 2t = 9 t = 4.5 Definition of equilateral ∆. Subtract 4y and add 6 to both sides. Divide both sides by 2. Thus JL = 2(4.5) + 1 = 10. Holt Geometry 4-8 Isosceles and Equilateral Triangles • Practice Problems – pp. 276-279 #12-24 even • Chapter 4 Review – pp. 284-285 #1-21, 28-30 • For #12-21, give the theorem/postulate that proves the triangles are congruent Holt Geometry