1. **State the problem:** We need to find the measure of angle $\angle G$ in triangle $FGH$ where the sides are given as $FH=15$, $HG=10$, and $FG=13$. 2. **Formula used:** To find an angle when all three sides are known, we use the Law of Cosines: $$\cos(\angle G) = \frac{FG^2 + HG^2 - FH^2}{2 \cdot FG \cdot HG}$$ 3. **Substitute the known values:** $$\cos(\angle G) = \frac{13^2 + 10^2 - 15^2}{2 \cdot 13 \cdot 10}$$ 4. **Calculate the squares:** $$\cos(\angle G) = \frac{169 + 100 - 225}{260}$$ 5. **Simplify the numerator:** $$\cos(\angle G) = \frac{44}{260}$$ 6. **Simplify the fraction by dividing numerator and denominator by 4:** $$\cos(\angle G) = \frac{\cancel{44}^{{11}}}{\cancel{260}^{{65}}}$$ 7. **Calculate the decimal value:** $$\cos(\angle G) \approx 0.1692$$ 8. **Find the angle by taking the inverse cosine:** $$\angle G = \cos^{-1}(0.1692) \approx 80.3^\circ$$ **Final answer:** $$m\angle G \approx 80.3^\circ$$