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## Explain why ABC is isosceles.

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**ABC and XAB are alternate interior angles formed by**XA, BC, and the transversal AB. Because XA || BC, ABCXAB. The diagram shows that XABACB. By the Transitive Property of Congruence, ABCACB. You can use the Converse of the Isosceles Triangle Theorem to conclude that ABAC. By the definition of an isosceles triangle, ABC is isosceles. Isosceles and Equilateral Triangles LESSON 4-5 Additional Examples Explain why ABC is isosceles. Quick Check**MOLNThe bisector of the vertex angle of an isosceles**triangle is the perpendicular bisector of the base. x = 90 Definition of perpendicular Isosceles and Equilateral Triangles LESSON 4-5 Additional Examples Suppose that mL = y. Find the values of x and y. mN = mLIsosceles Triangle Theorem mL = y Given mN = yTransitive Property of Equality mN + mNMO + mMON = 180 Triangle Angle-Sum Theorem y + y + 90 = 180 Substitute. 2y + 90 = 180 Simplify. 2y = 90 Subtract 90 from each side. y = 45 Divide each side by 2. Quick Check Therefore, x = 90 and y = 45.**Isosceles and Equilateral Triangles**LESSON 4-5 Additional Examples Suppose the raised garden bed is a regular hexagon. Suppose that a segment is drawn between the endpoints of the angle marked x. Find the angle measures of the triangle that is formed. Because the garden is a regular hexagon, the sides have equal length, so the triangle is isosceles. By the Isosceles Triangle Theorem, the unknown angles are congruent. Example 4 found that the measure of the angle marked x is 120°. The sum of the angle measures of a triangle is 180°. If you label each unknown angle y, 120 + y + y = 180. 120 + 2y = 180 2y = 60 y = 30 Quick Check So the angle measures in the triangle are 120°, 30° and 30°.