1. **State the problem:** We need to find the measure of the smallest angle in a triangle with sides 12, 23, and 34, where the smallest angle is opposite the side of length 12. 2. **Formula used:** Use the Law of Cosines to find the angle opposite side $a=12$: $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$ where $a=12$, $b=23$, and $c=34$. 3. **Calculate the cosine of angle $A$:** $$\cos A = \frac{23^2 + 34^2 - 12^2}{2 \times 23 \times 34} = \frac{529 + 1156 - 144}{2 \times 23 \times 34} = \frac{1541}{1564}$$ 4. **Simplify the fraction:** $$\cos A = \frac{\cancel{1541}}{\cancel{1564}}$$ (No common factors to cancel, so fraction remains $\frac{1541}{1564}$.) 5. **Find angle $A$ by taking the inverse cosine:** $$A = \cos^{-1}\left(\frac{1541}{1564}\right)$$ 6. **Calculate the value:** $$A \approx \cos^{-1}(0.9857) \approx 9.57^\circ$$ 7. **Conclusion:** The smallest angle of the triangle is approximately $9.57^\circ$ rounded to the nearest hundredth.