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Triangle Inequality Objective: – Students make conjectures about the measures of opposite sides and angles of triangles. If one side of a triangle is longer than the other sides, then its opposite angle is longer than the other two angles. Biggest Side is Opposite to Biggest Angle Medium Side is Opposite to Medium Angle Smallest Side is Opposite to Smallest Angle A m<B is greater than m<C C 9 4 6 B If one angle of a triangle is longer than the other angles, then its opposite side is longer than the other two sides. Converse is true also Biggest Angle Opposite ______ Medium Angle Opposite______ Smallest Angle Opposite______ A Angle B > Angle A > Angle C So AC >BC > AB 9 4 84◦ 47◦ C 6 B Triangle Inequality Objective: – determine whether the given triples are possible lengths of the sides of a triangle Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side A 9 C 4 6 B Inequalities in One Triangle They have to be able to reach!! 3 2 4 3 6 3 3 6 6 Triangle Inequality Theorem AB + AC > BC AB + BC > AC A AC + BC > AB 9 C 4 6 B Example: Determine if the following lengths are legs of triangles A) 4, 9, 5 B) 9, 5, 5 We choose the smallest two of the three sides and add them together. Comparing the sum to the third side: 4+5 ? 9 5+5 ? 9 9>9 10 > 9 Since the sum is not greater than the third side, this is not a triangle Since the sum is greater than the third side, this is a triangle Triangle Inequality Objective: – Solve for a range of possible lengths of a side of a triangle given the length of the other two sides. A triangle has side lengths of 6 and 12; what are the possible lengths of the third side? B 6 12 A X=? 1) 12 + 6 = 18 2) 12 – 6 = 6 Therefore: C 6 < X < 18 Examples Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. 5 in 12 in 10 yd. 23 yd. 18 ft. 12 ft.