1. Given the triangle with sides $a=8$, $b=10$, and $c=12$, we want to find angle $C$ opposite side $c$. 2. Use the Law of Cosines formula: $$c^2 = a^2 + b^2 - 2ab \cos C$$ 3. Substitute the known values: $$12^2 = 8^2 + 10^2 - 2 \times 8 \times 10 \times \cos C$$ 4. Calculate squares: $$144 = 64 + 100 - 160 \cos C$$ 5. Simplify the right side: $$144 = 164 - 160 \cos C$$ 6. Rearrange to isolate $\cos C$: $$144 - 164 = -160 \cos C$$ 7. Simplify left side: $$-20 = -160 \cos C$$ 8. Divide both sides by $-160$ showing cancellation: $$\frac{-20}{\cancel{-160}} = \frac{-160 \cos C}{\cancel{-160}}$$ 9. Simplify fraction: $$\cos C = 0.125$$ 10. Find angle $C$ by taking inverse cosine: $$C = \cos^{-1}(0.125)$$ 11. Calculate the value: $$C \approx 82.82^\circ$$ Final answer: Angle $C$ is approximately $82.82^\circ$.