4 9 Triangles 4 9 Isoscelesand Equilateral Triangles

4 -9 Triangles 4 -9 Isoscelesand Equilateral Triangles Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Geometry Holt

4 -9 Isosceles and Equilateral Triangles Objectives Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles. Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Vocabulary legs of an isosceles triangle vertex angle base angles Holt Mc. Dougal Geometry

4 -9 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 Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Reading Math The Isosceles Triangle Theorem is sometimes stated as “Base angles of an isosceles triangle are congruent. ” Holt Mc. Dougal Geometry

4 -9 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 Mc. Dougal Geometry

4 -9 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 Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Check It Out! Example 2 A Find m H = m G = x° Isosc. ∆ Thm. m H + m G + m F = 180 ∆ Sum Thm. Substitute the x + 48 = 180 given values. Simplify and subtract 2 x = 132 48 from both sides. x = 66 Divide both sides by 2. Thus m H = 66° Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Check It Out! Example 2 B Find m N. m P = m N Isosc. ∆ Thm. (8 y – 16) = 6 y 2 y = 16 y = 8 Substitute the given values. Subtract 6 y and add 16 to both sides. Divide both sides by 2. Thus m N = 6(8) = 48°. Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles The following corollary and its converse show the connection between equilateral triangles and equiangular triangles. Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Holt Mc. Dougal Geometry

4 -9 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 Mc. Dougal Geometry The measure of each of an equiangular ∆ is 60°. Subtract 32 both sides. Divide both sides by 2.

4 -9 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 Mc. Dougal Geometry Definition of equilateral ∆. Subtract 4 y and add 6 to both sides.

4 -9 Isosceles and Equilateral Triangles Check It Out! Example 3 Find the value of JL. ∆JKL is equiangular. Equiangular ∆ equilateral ∆ 4 t – 8 = 2 t + 1 2 t = 9 t = 4. 5 Definition of equilateral ∆. Subtract 4 y and add 6 to both sides. Divide both sides by 2. Thus JL = 2(4. 5) + 1 = 10. Holt Mc. Dougal Geometry

4 -9 Isosceles and Equilateral Triangles Lesson Quiz: Part I Find each angle measure. 1. m R 28° 2. m P 124° Find each value. 3. x 5. x Holt Mc. Dougal Geometry 20 4. y 26° 6

4 -9 Isosceles and Equilateral Triangles Lesson Quiz: Part II 6. The vertex angle of an isosceles triangle measures (a + 15)°, and one of the base angles measures 7 a°. Find a and each angle measure. a = 11; 26°; 77° Holt Mc. Dougal Geometry