1. GEOMETRY HELP Explain why ABC is isosceles. By the definition of an isosceles triangle, ABC is isosceles.  ABC and  XAB are alternate interior angles formed by XA, BC, and the transversal AB. Because XA || BC,  ABC  XAB. The diagram shows that  XAB  ACB. By the Transitive Property of Congruence,  ABC  ACB. You can use the Converse of the Isosceles Triangle Theorem to conclude that AB AC. Quick Check Isosceles and Equilateral Triangles LESSON 4-5 Additional Examples

2. GEOMETRY HELP Suppose that m  L = y. Find the values of x and y. m  N=m  LIsosceles Triangle Theorem m  L=yGiven m  N + m  NMO + m  MON=180Triangle Angle-Sum Theorem m  N=yTransitive Property of Equality y + y + 90=180Substitute. 2y + 90=180Simplify. 2y=90Subtract 90 from each side. y=45Divide each side by 2. Therefore, x = 90 and y = 45. MOLNThe bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. x=90Definition of perpendicular Isosceles and Equilateral Triangles LESSON 4-5 Additional Examples Quick Check

3. GEOMETRY HELP 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 So the angle measures in the triangle are 120°, 30° and 30°. 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. Isosceles and Equilateral Triangles LESSON 4-5 Additional Examples Quick Check