1. The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.

2. Example 1: Ordering Triangle Side Lengths and Angle Measures
Write the angles in order from smallest to largest.
The shortest side is , so the smallest angle is F.
The angles from smallest to largest are F, H and G.

3. Example 2: Ordering Triangle Side Lengths and Angle Measures
Write the sides in order from shortest to longest.
mR = 180° – (60° + 72°) = 48°
The smallest angle is R, so the shortest side is .
48°
The sides from shortest to longest are

4. Example 3:

5. Example 4:

6. A triangle is formed by three segments, but not every set of three segments can form a triangle.

7. A certain relationship must exist among the lengths of three segments in order for them to form a triangle.
NOTE: Just check that the sum of the two shorter sides is greater than the longest side.

8. Example 5: Applying the Triangle Inequality Theorem

9. Example 5: Applying the Triangle Inequality Theorem

10. Example 6: Finding Possible Side Lengths
The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.
Let x represent the length of the third side. Then apply the Triangle Inequality Theorem.
x + 8 > 13
> x
x > 5
21 > x
Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches.

11. Example 7
The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side.

12. You can also use side lengths to classify a triangle as acute or obtuse.