1. **State the problem:** We have a right triangle with vertices A, B, and C, where the right angle is at C. The side opposite angle C (AC) is 6 units, the side opposite angle A (AB) is 8 units, and we need to find the measure of angle A in degrees. 2. **Recall the relevant formula:** In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. Here, angle A is adjacent to side AC (6 units) and opposite side BC (unknown), with hypotenuse AB (8 units). 3. **Apply the cosine formula:** $$\cos(A) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{6}{8} = 0.75$$ 4. **Find angle A:** Use the inverse cosine function to find angle A: $$A = \cos^{-1}(0.75)$$ 5. **Calculate the value:** $$A \approx 41.41^\circ$$ 6. **Round the answer:** The angle A rounded to the nearest hundredth is: $$41.41^\circ$$ **Final answer:** $\boxed{41.41^\circ}$