Geometry: Triangle Inequality

On this page we hope to clear up any problems that you might have with triangle inequality. Read on or follow any of the links below to start understanding triangle inequality better!

Triangle Inequality Theorem
Inequalities for two triangles
Quiz on triangle inequality

Triangle Inequality Theorem

Triangle Inequality Theorem
The sum of the measures of any two sides of any triangle is greater than the measure of the third side.

In English, that means that in a triangle, you can pick any two sides' measures, and when you add them together, the sum will be greater than the measure of the third side.

Example:

1.  Problem: Can a triangle have the following measures?
             3, 10, and 8
             9, 17, and 8
   Solution: The first set of measures can form a triangle because
             3 + 10 is greater than 8, 
             8 + 10 is greater than 3, and
             3 + 8 is greater than 10.

             The second set of measures cannot form a triangle be-
             cause 8 + 9 is equal to 17.

Inequalities for Two Triangles

The two theorems that apply to inequalities in two triangles are explained below (and illustrated in the accompanying figure).

1. If two triangles have two sides that are congruent, the triangle with the larger third side will have a larger included angle.
2. If two triangles have two sides that are congruent, the triangle with the larger included angle will have a larger third side.

Take the quiz on triangle inequality. The quiz is very useful for either review or to see if you've really got the topic down.

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