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LESSON 4-5 NOTES: ISOSCELES AND EQUILATERAL TRIANGLES Isosceles Triangle: A triangle with two congruent sides. X Vertex Angle 1 2 Leg Leg Base Y Base Angles Z B Diagram for proof of Theorem 4-3 Theorem 4-3 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Proof of Theorem 4-3: Refer to the diagram of isosceles triangle XYZ above. Given: , bisects ∠YXZ Prove: STATEMENTS ∠Y ∠Z REASONS 1) 1) 2) 2) 3) 3) 4) 4) 5) 5) 6) 6) EXAMPLES/PRACTICE: Find the value of a, b and c in each isosceles triangle. 1) 2) 3) a a c a c a 50 a a b 110 a b ac a a a b a 75 EXAMPLES/PRACTICE: Find the value of x and the measure of each unknown angle in each isosceles triangle. 40 3) 4) 5) 6x − 4 65 9x + 7 8x − 5 5(x + 5) Remember that a converse of a statement "switches" the hypothesis ("if" part) and the conclusion ("then" part) of the statement. Theorem 4-4 Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Theorem 4-5 If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base. bisects the vertex angle ∠YXZ. We proved earlier by SAS that ∆XYB ∆XZB. So by CPCTC, ∠XBY ∠XBZ and . Proof of Theorem 4-5: X ∠XBY and ∠XBZ form a linear pair. By the Linear Pair Postulate, m∠XBY + m∠XBZ = 180. Because ∠XBY ∠XBZ, both angles are right angles. 1 2 Because Y B Z , B is the midpoint of . So , which bisects the vertex angle, ∠YXZ, is the perpendicular bisector of the base, , of isosceles triangle YXZ, A corollary is a theorem that can be proved easily using another theorem. Since a corollary is a theorem, you can use it as a reason in a proof. Corollary to Theorem 4-3 If a triangle is equilateral, then the triangle is equiangular. If: then: Corollary to Theorem 4-4 If a triangle is equiangular, then the triangle is equilateral. If: then: Because the sum of the interior angles of a triangle is 180 degrees, each angle of an equiangular/equilateral triangle is 60 degrees. EXAMPLES/PRACTICE: Find the values of x and y in each. 1) 2) x 60 y 8x - 12 y