1. **State the problem:** Given a triangle with sides $a=15$, $b=18$, and $c=20$, find the measure of angle $C$ opposite side $c$ using the Law of Cosines. 2. **Formula:** The Law of Cosines states: $$c^2 = a^2 + b^2 - 2ab \cos C$$ This formula relates the lengths of the sides of a triangle to the cosine of one of its angles. 3. **Rearrange to solve for $\cos C$:** $$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$ 4. **Substitute the given values:** $$\cos C = \frac{15^2 + 18^2 - 20^2}{2 \times 15 \times 18} = \frac{225 + 324 - 400}{540} = \frac{149}{540}$$ 5. **Calculate $\cos C$:** $$\cos C = \frac{149}{540} \approx 0.2759$$ 6. **Find angle $C$ by taking the inverse cosine:** $$C = \cos^{-1}(0.2759) \approx 74.0^\circ$$ **Final answer:** $$\boxed{74.0^\circ}$$