## Triangle Inequalities: Sides and Angles

You have just seen that if a triangle has *equal sides*, the angles opposite these sides are equal, and if a triangle has *equal angles*, the sides opposite these angles are equal. There are two important theorems involving unequal sides and unequal angles in triangles. They are:

*Theorem 36:* If two sides of a triangle are unequal, then the measures of the angles opposite these sides are unequal, and the greater angle is opposite the greater side.

*Theorem 37:* If two angles of a triangle are unequal, then the measures of the sides opposite these angles are also unequal, and the longer side is opposite the greater angle.

**Example 1:** Figure 1 shows a triangle with angles of different measures. List the sides of this triangle in order from least to greatest.

**Figure 1 **List the sides of this triangle in increasing order.

Because 30° < 50° < 100°, then *RS* < *QR* < *QS*.

**Example 2:** Figure 2 shows a triangle with sides of different measures. List the angles of this triangle in order from least to greatest.

**Figure 2 **List the angles of this triangle in increasing order.

Because 6 < 8 < 11, then *m* ∠ *N* < *m* ∠ *M* < *m* ∠ *P*.

**Example 3:** Figure 3 shows right Δ *ABC*. Which side must be the longest?

**Figure 3 **Identify the longest side of this right triangle.

Because ∠ *A* + *m* ∠ *B* + *m* ∠ *C* = 180 ° (by Theorem 25) and *m* ∠ = 90°, we have *m* ∠ *A* + *m* ∠ *C* = 90°. Thus, each of *m* ∠ *A* and *m* ∠ *C* is less than 90°. Thus ∠ *B* is the angle of greatest measure in the triangle, so its opposite side is the longest. Therefore, the hypotenuse, *AC* , is the longest side in a right triangle.