1. **State the problem:** We want to find the angle $C$ in a triangle using the Law of Cosines. 2. **Formula:** The Law of Cosines states: $$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$$ where $a$, $b$, and $c$ are the sides of the triangle opposite angles $A$, $B$, and $C$ respectively. 3. **Given values:** $a=8$, $b=10$, $c=12$. 4. **Substitute values into the formula:** $$\cos C = \frac{8^2 + 10^2 - 12^2}{2 \times 8 \times 10}$$ 5. **Calculate the squares:** $$\cos C = \frac{64 + 100 - 144}{160}$$ 6. **Simplify numerator:** $$\cos C = \frac{20}{160}$$ 7. **Simplify the fraction by dividing numerator and denominator by 20:** $$\cos C = \frac{\cancel{20}}{\cancel{160}} = \frac{1}{8} = 0.125$$ 8. **Find angle $C$ by taking the inverse cosine:** $$C = \cos^{-1}(0.125)$$ 9. **Calculate the angle:** $$C \approx 82.82^\circ$$ **Final answer:** $$C \approx 82.82^\circ$$