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# Triangle Inequality Theorem

## What is the Triangle Inequality Theorem?

**The sum of two sides of a triangle must be greater than the third side.**

This theorem can be used to prove if a combination of three triangle side lengths is possible. See the image below for an illustration of the triangle inequality theorem.

### Example Problem

Referencing sides *x*, *y*, and *z* in the image above, use the triangle inequality theorem to eliminate impossible triangle side length combinations from the following list.

1.) x = 2, y = 3, z = 5

2.) x = 5, y = 12, z = 13

3.) x = 3, y = 4, z = 5

4.) x = 12, y = 13, z = 27

5.) x = 2, y = 9, z = 12

Solution:**Side length combinations #1, #4, and #5 do not satisfy the requirements of the triangle inequality theorem and therefore are not possible.**