Triangle Inequality

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 9 4 B 6 C

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◦ B 47◦ 6 C

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 4 B 6 C

Inequalities in One Triangle • They have to be able to reach!! 3 2 4 3 6 6 3 3 6

Triangle Inequality Theorem • AB + BC > AC • AB + AC > BC A • AC + BC > AB 9 4 B 6 C

Example: Determine if the following lengths are legs of triangles B) 9, 5, 5 A) 4, 9, 5 We choose the smallest two of the three sides and add them together. Comparing the sum to the third side: 4 + 5 ? 9 9 > 9 5 + 5 ? 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.

6 12 X = ? A triangle has side lengths of 6 and 12; what are the possible lengths of the third side? 1) 12 + 6 = 18 2) 12 – 6 = 6 6 < X < 18 Therefore:

Examples Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. 10 yd. 23 yd. 18 ft. 12 ft. 5 in 12 in

Triangle Inequality