1. Warm-up Solve: 1) 2x + 1+4x +4x-11= 180 Compare greater than >, less than < or equal = 4+5___ 9 5+5__ 9 Find a number x. 6<x<18

2. Solution: 1) 10x -10=180 10x=180+10 10x=190 x=190/10 x=19 4+5=9 5+5>9 6<x<18 x=7, x=16 every number greater than 6 and less than 18.

3. Essential Questions: 1-What does the sum of two sides of a triangle tell me about the third side? 2- In a triangle, what is the relationship of an angle of a triangle and its opposite side? 3-What is the relationship between the exterior angle of a triangle and either of its remote interior angles? 4- For two triangles with two pairs of congruent corresponding sides, what is the

4. Essential Questions: Relationship between the included angle and the third side? 5- How do I use the Converse of the Pythagorean Theorem to determine if a triangle is a right triangle?

5. Vocabulary 1. Triangle inequality 2. Side-angle inequality 3. 3. Side-side-side inequality 4. Exterior angle inequality

6. Inequalities in One Triangle 6 3 2 6 3 3 4 3 6 Note that there is only one situation that you can have a triangle; when the sum of two sides of the triangle are greater than the third. They have to be able to reach!!

7. Triangle Inequality Theorem AB + BC > AC A B C AB + AC > BC AC + BC > AB

8. Triangle Inequality Theorem A B C Biggest Side Opposite Biggest Angle Medium Side Opposite Medium Angle Smallest Side Opposite Smallest Angle 3 5 m<B is greater than m<C

9. Triangle Inequality Theorem Converse is true also Biggest Angle Opposite _____________ Medium Angle Opposite ______________ Smallest Angle Opposite _______________ B C A 65 30 Angle A > Angle B > Angle C So CB >AC > AB

10. Example: List the measures of the sides of the triangle, in order of least to greatest. 10x - 10 = 180 Solving for x: Therefore, BC < AB < AC <A = 2x + 1 <B = 4x <C = 4x -11 2x +1 + 4x + 4x - 11 =180 10x = 190 X = 19 Plugging back into our Angles: <A = 39 o ; <B = 76; <C = 65 Note: Picture is not to scale

11. Using the Exterior Angle Inequality Example: Solve the inequality if AB + AC > BC x + 3 x + 2 A B C (x+3) + (x+ 2) > 3x - 2 3x - 22x + 5 > 3x - 2 x < 7

12. 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

13. Example: a triangle has side lengths of 6 and 12; what are the possible lengths of the third side? 6 12 X = ? 12 + 6 = 18 12 – 6 = 6 Therefore: 6 < X < 18

14. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side