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6-8 The Triangle-Sum Property Objective: 1. 2. use the triangle-sum property to find measures of given angles explain the consequences of the triangle-sum property Big Idea: In any triangle, the sum of the measures of the three angles is 180 Triangle-Sum Property – In any triangle, the sum of the measures of the angles is 180. Deductive Reasoning – (deduction) means that you reason logically from some things that you know are true or assume to be true. Finding the measure of a 3rd angle of a triangle. 69 47 x 47 + 69 +x = 180 116 + x = 180 116 + -116 + x = 180 + -116 x = 64 Check: 47 + 69 + 64 = 180 180 = 180 Angle exterior to a triangle – an angle that is outside a triangle, made by extending one of the sides of the triangle. It is a linear pair with the interior (inside) angle of the triangle. Finding the measure of an angle exterior to a triangle Angle exterior to Δ 31 62 y First find y, the inside angle 62 + 31 + y = 180 93 + y = 180 93 + -93 + y = 180 + -93 y = 87 x Then find x by using linear pair relationship with y 87 + x = 180 87 + -87 + x = 180 + -87 x = 93 Finding the measures of angles given triangles and parallel lines. 4 = 68 so 2 = 68 by alt. interior 68 10 9 7 8 11 4 1 5 2 6 3 1 + 2 + 90 = 180 1 + 68 + 90 = 180 1 + 158 = 180 1 + 158 + -158 = 180 + -158 1 = 22 Find all the measures of ’s v, w, x, y, z Extend the lines to see parallel line angle relationships. C B X A V Y 36º W 119º E mV = 83º Z D mW = 61º because it’s a linear pair with BED mW =61º mX =83º mY =97º mZ =61º mX – use triangle sum theorem with A and BEA 36º + 61º + X = 180º -97 + 97 + X = 180 + -97 X = 83º mY = 97º because it is a linear pair with EBA mZ = 61º because it is a corresponding angle with AEB mV = 83º because it is a corresponding angle with ABE