1. Inequalities in Triangles

2. Definition of Inequality
For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c.

3. Exterior Angle Inequality
The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles.

4. Examples
Find all measures less than angle 7.

5. Examples
Find all measures greater than angle 6.

6. Examples
Find all measures greater than angle 1.

7. Examples
Find all measures greater than angle 8.

8. Angle-Side Relationships
If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.

9. Examples
List the angles of PQR in order from smallest to largest.

10. Examples
List the angles of PQR in order from smallest to largest.

11. Angle-Side Relationships
If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

12. Examples
List the sides of FGH in order from shortest to longest.

13. Examples
List the sides of WXY in order from shortest to longest.

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

15. Examples
Is it possible to form a triangle with the given side lengths? If not, explain why not.
8, 15, 17

16. Examples
Is it possible to form a triangle with the given side lengths? If not, explain why not.
6, 8, 14

17. Examples
Is it possible to form a triangle with the given side lengths? If not, explain why not.
15, 16, 30

18. Examples
Is it possible to form a triangle with the given side lengths? If not, explain why not.
2, 8, 11

19. Triangle Inequality Theorem
When the lengths of two sides of a triangle are known, the third side can be any length in a range of values. You can use the Triangle Inequality Theorem to determine the range of possible lengths for the third side.

20. Triangle Inequality Theorem
To do this, find the sum and the difference of the two known sides. The sum provides the upper limit and the difference provides the lower limit.
= 22
13 – 9 = 4
4 < n < 22