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Angle and Side Length Relationships Complementary Angles Angles in which the sum of their measures is 90o. Supplementary Angles Angles in which the sum of their measures is 180o. Congruent Angles Angles that have the same measure. Example Solve for x if angle A and angle B are complementary. (8x – 15)o (3x – 5)o Since the two angles are complementary, the sum of their measures should equal 90o. Steps 1. Set sum of both angles = 90 2. Simplify both sides 3. Add 20 to both sides 4. Divide both sides by 11 3x – 5 + 8x – 15 = 90 11x – 20 = 90 11x = 110 x = 10 Angle and Side Relationship in One Triangle The largest angle opens to the longest side The smallest angle opens to the shortest side Example B 45o 15 6 115o C A 30o 11 © LaurusSoft, Inc. Example Find the perimeter of ∆YXZ given ∆ABC B Z 12 m 10 m C ~ ∆YXZ. 4m Y X 8m A Answer: Side AC of the blue triangle corresponds to side YZ of the green triangle. Since it is given that the triangles are similar and the length of AC is twice the length of the YZ, we can conclude that the lengths of each of the sides of the blue triangle is twice the length of its corresponding green sides. Since AB corresponds with YX, we can conclude that YX has a length of 5 m (half of 10 m). Since BC corresponds with XZ, we can conclude that XZ has a length of 6 m (half of 12 m). To find the perimeter, we add the lengths of the three sides together (4 + 5 + 6). So, the perimeter of ∆YXZ is 15 m. Interior and Exterior Angles interior angle exterior angle Below is a list of formulas that relate the interior and exterior angles to the number of sides a polygon has. Sum of Interior Angles Sum of the measures of the interior angles of a convex polygon with n sides = (n – 2)180o. Sum of Exterior Angles Sum of the measures of the exterior angles of a convex polygon with n sides = 360o. © LaurusSoft, Inc. Example n: 5 Interior sum: (5 – 2)180 = 540o Exterior sum: 360o Interior Angle of Regular Polygon Interior Angle Measure a= 180(n − 2) n Example Find the measure of the interior angle of a regular hexagon. Answer: Since a hexagon has 6 sides, substitute 6 in for n in the formula 180(n − 2) 180(6 − 2) 180(4) 720 = = = = 120 . n 6 6 6 So, the measure of an interior angle of a regular hexagon is 120o. © LaurusSoft, Inc.