In Δ *TAB* (Figure ), if *T, A*, and *B* represent three points on a map and you want to go from *T* to *B,* going from *T* to *A* to *B* would obviously be longer than going directly from *T* to *B.* The following theorem expresses this idea.

**Figure 1 **Two paths from T to B.

*Theorem 38 (Triangle Inequality Theorem):* The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

**Example 1:** In Figure 2, the measures of two sides of a triangle are 7 and 12. Find the range of possibilities for the third side.

**Figure 2 **What values of x will make a triangle possible?

Using the *Triangle Inequality Theorem,* you can write the following:

7 + *x* > 12, so *x* > 5

7 + 12 > *x*, so 19 > *x* (or *x* < 19)

Therefore, the third side must be more than 5 and less than 19.