1. **Problem statement:** Find the measure of angle $C$ in triangle $ABC$ with sides $AB=8$, $BC=7$, and $AC=13$. The angle to find is at vertex $C$. 2. **Formula used:** Use the Law of Cosines to find an angle when all three sides are known: $$\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$$ where $a$ and $b$ are the sides adjacent to angle $C$, and $c$ is the side opposite angle $C$. 3. **Identify sides:** Here, angle $C$ is between sides $BC=7$ and $AC=13$, so $a=7$, $b=13$, and the side opposite angle $C$ is $AB=8$, so $c=8$. 4. **Apply Law of Cosines:** $$\cos(C) = \frac{7^2 + 13^2 - 8^2}{2 \times 7 \times 13} = \frac{49 + 169 - 64}{182} = \frac{154}{182}$$ 5. **Simplify fraction:** $$\frac{154}{182} = \frac{\cancel{14} \times 11}{\cancel{14} \times 13} = \frac{11}{13}$$ 6. **Calculate angle $C$:** $$C = \cos^{-1}\left(\frac{11}{13}\right)$$ Using a calculator, $$C \approx 32.24^\circ$$ 7. **Round to nearest degree:** $$m\angle C \approx 32^\circ$$ **Final answer:** The measure of angle $C$ is approximately $32^\circ$.