Inequalities in One Triangle

Comparing Sides and Angles

• If one side of a Triangle is longer than another side, then the Angle opposite the longer side is larger than the Angle opposite the shorter side.
• If one Angle of a Triangle is larger than another Angle, then the side opposite the larger Angle is longer than the side opposite the smaller Angle.

Exterior Angle Inequality Theorem

The measure of an exterior Angle of a Triangle is greater than the measure of either of the two non-adjacent (remote) interior angles.

Ordering Sides and Angles

In any Triangle, you can order the sides from shortest to longest and the angles from smallest to largest — these orderings match:

• Shortest side ↔ Smallest Angle
• Medium side ↔ Medium Angle
• Longest side ↔ Largest Angle

Example: In △ABC with AB = 5, BC = 8, AC = 6, the angles from smallest to largest are: ∠C < ∠B < ∠A (Angle opposite the shortest side is smallest).