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Objectives: Use and Apply Properties of isosceles triangles

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.

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Objectives: Use and Apply Properties of isosceles triangles

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

  2. Isosceles Triangle Theorem

  3. It is given that XYXZ. By the Reflexive Property of Congruence, XBXB. Triangle Proofs Examine the diagram below. Suppose that you draw XBYZ. Can you use SAS to prove XYBXZB? 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.

  4. MOLNThe bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. x = 90 Definition of perpendicular Triangle Proofs Suppose that mL = y. Find the values of x and y. mN = mLIsosceles Triangle Theorem mL = y Given mN = yTransitive Property of Equality mN + mNMO + mMON = 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. Therefore, x = 90 and y = 45.

  5. Triangle Proofs 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. 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.

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