1. **State the problem:** We need to find the measure of angle $Y$ in triangle $XYZ$ where the sides are $XY=12$, $XZ=15$, and $YZ=20$. 2. **Formula used:** Use the Law of Cosines to find angle $Y$. The Law of Cosines states: $$\cos Y = \frac{XY^2 + YZ^2 - XZ^2}{2 \cdot XY \cdot YZ}$$ 3. **Substitute the known values:** $$\cos Y = \frac{12^2 + 20^2 - 15^2}{2 \cdot 12 \cdot 20} = \frac{144 + 400 - 225}{480} = \frac{319}{480}$$ 4. **Calculate the cosine value:** $$\cos Y = 0.6645833...$$ 5. **Find angle $Y$ by taking the inverse cosine:** $$Y = \cos^{-1}(0.6645833)$$ 6. **Calculate the angle in degrees:** $$Y \approx 48.3^\circ$$ **Final answer:** $$m\angle Y \approx 48.3^\circ$$