1. **State the problem:** We need to find the measure of angle $\angle X$ in triangle $WYX$ where the sides are $WY=12$, $XY=13$, and $WX=15$. 2. **Formula used:** To find an angle when all three sides are known, we use the Law of Cosines: $$\cos(\angle X) = \frac{WY^2 + WX^2 - XY^2}{2 \cdot WY \cdot WX}$$ 3. **Substitute the known values:** $$\cos(\angle X) = \frac{12^2 + 15^2 - 13^2}{2 \cdot 12 \cdot 15}$$ 4. **Calculate the squares:** $$\cos(\angle X) = \frac{144 + 225 - 169}{360}$$ 5. **Simplify the numerator:** $$\cos(\angle X) = \frac{200}{360}$$ 6. **Simplify the fraction:** $$\cos(\angle X) = \frac{\cancel{200}^{\times 1}}{\cancel{360}^{\times 1.8}} = \frac{5}{9} \approx 0.5556$$ 7. **Find the angle using inverse cosine:** $$\angle X = \cos^{-1}(0.5556) \approx 56.25^\circ$$ 8. **Round to the nearest tenth:** $$\angle X \approx 56.3^\circ$$ **Final answer:** $m\angle X = 56.3^\circ$