What is the Triangle Inequality Theorem?

The triangle inequality theorem tells us that:

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.

Triangle Inequality Theorem Example Problem

Referencing sides x, y, and z in the image from the above section, 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.