How to use and apply properties of isosceles triangles. Chapter 4.5GeometryStandard/Goal: 4.1

1. Check and discuss assignment from yesterday. 2. Work on Quiz Read, write, and discuss how to use and apply properties of isosceles triangles. 4. Work on assignment.

Isosceles triangle legs Isosceles triangle base Base angles Vertex angle

If two sides of a triangle are congruent, Then the angles opposite those sides are congruent.

If two angles of a triangle are congruent, Then the sides opposite the angles are congruent.

The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

corollary is a statement that follows immediately from a theorem.

If a triangle is equilateral, Then the triangle is equiangular.

If a triangle is equiangular, Then the triangle is equilateral.

Explain why ABC is isosceles. By the definition of an isosceles triangle, ABC is isosceles. Lesson 4-5  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.

Suppose that m  L = y. Find the values of x and y. m  N = m  L Isosceles Triangle Theorem m  L=y Given m  N + m  NMO + m  MON =180Triangle Angle-Sum Theorem m  N = y Transitive Property of Equality y + y + 90=180Substitute. 2 y + 90=180Simplify. 2 y =90Subtract 90 from each side. y =45Divide each side by 2. Therefore, x = 90 and y = 45. MOLN The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. x =90Definition of perpendicular Lesson 4-5

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, y + y = y =180 2 y =60 y =30 So the angle measures in the triangle are 120, 30 and 30. Lesson 4-5 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.

Kennedy, D., Charles, R., Hall, B., Bass, L., Johnson, A. (2009) Geometry Prentice Hall Mathematics. Power Point made by: Robert Orloski Jerome High School.