1. Triangle Inequality Objective: –Students make conjectures about the measures of opposite sides and angles of triangles.

2. If one side of a triangle is longer than the other sides, then its opposite angle is longer than the other two angles. A B C  Biggest Side is Opposite to Biggest Angle  Medium Side is Opposite to Medium Angle  Smallest Side is Opposite to Smallest Angle 4 9 m<B is greater than m<C 6

3. Converse is true also  Biggest Angle Opposite ______  Medium Angle Opposite______  Smallest Angle Opposite______ Angle B > Angle A > Angle C So AC >BC > AB If one angle of a triangle is longer than the other angles, then its opposite side is longer than the other two sides. A B C 4 9 6 84 ◦ 47 ◦

4. Triangle Inequality Objective: –determine whether the given triples are possible lengths of the sides of a triangle

5. A B C 4 9 6 Triangle Inequality Theorem  The sum of the lengths of any two sides of a triangle is greater than the length of the third side

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

7. Triangle Inequality Theorem  AB + BC > AC  AB + AC > BC  AC + BC > AB A B C 4 9 6

8. Example: Determine if the following lengths are legs of triangles A)4, 9, 5 4 + 5 ? 9 9 > 9 We choose the smallest two of the three sides and add them together. Comparing the sum to the third side: B) 9, 5, 5 Since the sum is not greater than the third side, this is not a triangle 5 + 5 ? 9 10 > 9 Since the sum is greater than the third side, this is a triangle

9. Triangle Inequality Objective: –Solve for a range of possible lengths of a side of a triangle given the length of the other two sides.

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

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