## Geometry: Common Core (15th Edition)

$\overline{TU} < \overline{UV} < \overline{TV}$
First, let's figure out the measures of all the angles in the triangle given. We already have the measures of two of the angles. One angle is $30^{\circ}$ and the other is $90^{\circ}$ because it is a right angle. We use the triangle-sum theorem to figure out the measure of the third angle: $m \angle T = 180 - (90 + 30)$ Evaluate what is in parentheses first, according to order of operations: $m \angle H = 180 - (120)$ Subtract to solve: $m \angle H = 60$ Let's now put the measures of the angles in order from smallest to largest: $30^{\circ} < 60^{\circ} < 90^{\circ}$ Now, put the angles corresponding to these measures in order from smallest to largest: $\angle V < \angle T < \angle U$ From Theorem 5-11, which states that the longer side of a triangle is opposite the largest angle, we can write out the order of sides from shortest to longest: $\overline{TU} < \overline{UV} < \overline{TV}$