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Transcript
Unit 5 – Congruent Triangles Lesson 5 - Isosceles and Equilateral Triangles (p. 210) The Isosceles Triangle Theorems Legs of an isosceles triangle - __________________________________________________ Base of an isosceles triangle - __________________________________________________ Vertex angle of an isosceles triangle - ___________________________________________ Base angles of an isosceles triangle - ____________________________________________ Theorem 4-3 (Isosceles Triangle Theorem) – If two sides of a triangle are congruent, then the angles opposite those sides are congruent. A B Theorem 4-4 (Converse of Isosceles Triangle Theorem) – If two angles of a triangle are congruent, then the sides opposite the angles are congruent. AC BC Theorem 4-5 – The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. 1. CD bisects ACB (cuts the vertex angle into two congruent angles) 2. CD AB (creates right angles with the base) 3. CD bisects AB (cuts the base into two congruent segments) - Proving the Isosceles Triangle Theorem a. Developing Proof To prove the Isosceles Triangle Theorem, begin with isosceles XYZ with XY XZ . Draw XB , the bisector of the vertex angle YXZ . Given: XY XZ , XB bisects YXZ . Prove: Y Z Lesson 5 – Isosceles and Equilateral Triangles Page 1 Revised Fair 2014-2015 Unit 5 – Congruent Triangles - Using the Isosceles Triangle Theorem a. Developing Proof Explain why RST is isosceles. Explain. b. Can you conclude that RUV is isosceles? Explain. - Using Algebra a. Find the values of x and y. b. Suppose mL 43 . Find the values of x and y. Corollary - ________________________________________________________________ Corollary to Theorem 4-3 – If a triangle is equilateral, then the triangle is equiangular. X Y Z Corollary to Theorem 4-4 – If a triangle is equiangular, then the triangle is equilateral. XY YZ ZX - Real-World Connection a. Landscaping A landscaper uses rectangles and equilateral triangles for the path around the hexagonal garden. Find the value of x. b. What is the measure of the angle at each outside corner of the path? Lesson 5 – Isosceles and Equilateral Triangles Page 2 Revised Fair 2014-2015