1. **State the problem:** We have a right triangle UTS with a right angle at vertex U. The side UT is 56 units, and the side US is $28\sqrt{2}$ units. We need to find the measure of angle $\angle T$. 2. **Identify the sides relative to angle $\angle T$:** - Side opposite $\angle T$ is US = $28\sqrt{2}$. - Side adjacent to $\angle T$ is UT = 56. 3. **Use the tangent ratio:** The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. $$\tan(\angle T) = \frac{\text{opposite}}{\text{adjacent}} = \frac{US}{UT} = \frac{28\sqrt{2}}{56}$$ 4. **Simplify the fraction:** $$\frac{28\sqrt{2}}{56} = \frac{\cancel{28}\sqrt{2}}{\cancel{56} \times 2} = \frac{\sqrt{2}}{2}$$ 5. **Find the angle $\angle T$:** We know that $$\tan(\angle T) = \frac{\sqrt{2}}{2}$$ Using the inverse tangent function, $$\angle T = \tan^{-1}\left(\frac{\sqrt{2}}{2}\right)$$ 6. **Evaluate the angle:** $$\tan^{-1}\left(\frac{\sqrt{2}}{2}\right) = 35^\circ$$ (approximately) **Final answer:** $$m \angle T = 35^\circ$$