Section 4-5 Isosceles and Equilateral Triangles SPI 32C: determine congruence or similarity between triangles SPI 32M: justify triangle congruence given a diagram Objectives: Use and Apply Properties of isosceles triangles Isosceles Triangle

Isosceles Triangle Theorem

Examine the diagram below. Suppose that you draw XB YZ. Can you use SAS to prove XYB XZB? Explain. By the definition of perpendicular,  XBY =  XBZ. However, because the congruent angles are not included between the congruent corresponding sides, the SAS Postulate does not apply. You cannot prove the triangles congruent using SAS. It is given that XY XZ. By the Reflexive Property of Congruence, XB XB. Triangle Proofs

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 Triangle Proofs

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. 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 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. Triangle Proofs