1. **State the problem:** We need to find the measure of the largest angle in a triangle with sides 12, 23, and 34. The largest angle is opposite the longest side, which is 34. 2. **Formula used:** Use the Law of Cosines to find the angle opposite side $c=34$: $$\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$$ where $a=12$, $b=23$, and $c=34$. 3. **Calculate the cosine of the angle:** $$\cos(C) = \frac{12^2 + 23^2 - 34^2}{2 \times 12 \times 23} = \frac{144 + 529 - 1156}{552} = \frac{673 - 1156}{552} = \frac{-483}{552}$$ 4. **Simplify the fraction:** $$\cos(C) = \frac{\cancel{-483}}{\cancel{552}}$$ (No common factors to cancel, so fraction remains $-\frac{483}{552}$.) 5. **Calculate the angle $C$:** $$C = \cos^{-1}\left(-\frac{483}{552}\right)$$ Using a calculator, $$C \approx \cos^{-1}(-0.875) \approx 151.04^\circ$$ 6. **Answer:** The largest angle measures approximately **151.04 degrees** rounded to the nearest hundredth.