1. **Problem Statement:** We are given a right triangle with vertices B, C, and A. The right angle is at vertex C. Side BC measures 6, side BA measures 7, and side CA (opposite angle A) is unknown. We need to find the measure of angle $\angle A$ in degrees, rounded to the nearest hundredth. 2. **Identify the sides:** In a right triangle, the side opposite the right angle is the hypotenuse. Here, side BA is the hypotenuse with length 7. 3. **Use the trigonometric definition:** To find $\angle A$, we can use the cosine function, which relates the adjacent side to the hypotenuse: $$\cos(\angle A) = \frac{\text{adjacent side}}{\text{hypotenuse}}$$ The side adjacent to $\angle A$ is BC, which measures 6. 4. **Calculate cosine of $\angle A$:** $$\cos(\angle A) = \frac{6}{7}$$ 5. **Find $\angle A$ using inverse cosine:** $$\angle A = \cos^{-1}\left(\frac{6}{7}\right)$$ 6. **Calculate the value:** $$\angle A \approx \cos^{-1}(0.857142857) \approx 30.96^\circ$$ 7. **Final answer:** $\boxed{30.96^\circ}$