1. **Problem:** Find the angle measure $C$ given $\cos C = 0.9063$.\n\n2. **Formula:** Use the inverse cosine function to find the angle: $$C = \cos^{-1}(0.9063)$$\n\n3. **Calculation:** Using a calculator, find $$C = \cos^{-1}(0.9063) \approx 25^\circ$$\n\n4. **Explanation:** The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. To find the angle from the cosine value, we use the inverse cosine function.\n\n**Final answer:** $C \approx 25^\circ$