1. **Problem Statement:** We are given a right triangle with vertices A, B, and C, where the right angle is at vertex C. The side opposite vertex A is 5 units, the side opposite vertex C (one leg) is 7 units, and the hypotenuse is unknown. We need to find the measure of angle $\angle A$ in degrees, rounded to the nearest hundredth. 2. **Relevant Formula:** In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse. We can use the Pythagorean theorem to find the hypotenuse first: $$c = \sqrt{a^2 + b^2}$$ where $a$ and $b$ are the legs, and $c$ is the hypotenuse. 3. **Calculate the hypotenuse:** $$c = \sqrt{5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74} \approx 8.6023$$ 4. **Calculate $\angle A$ using sine:** $$\sin(\angle A) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{5}{8.6023} \approx 0.5812$$ 5. **Find the angle in degrees:** $$\angle A = \arcsin(0.5812) \approx 35.54^\circ$$ 6. **Final answer:** $$\boxed{35.54^\circ}$$ This is the measure of angle $\angle A$ rounded to the nearest hundredth.