1. **State the problem:** We need to find the measure of angle $\angle V$ in triangle $VWX$ where the sides are $VW=5$, $VX=7$, and $XW=6$. 2. **Formula used:** To find an angle when all three sides are known, use the Law of Cosines: $$\cos(\angle V) = \frac{a^2 + b^2 - c^2}{2ab}$$ where $a$ and $b$ are the sides adjacent to the angle, and $c$ is the side opposite the angle. 3. **Identify sides:** For $\angle V$, the sides adjacent are $VW=5$ and $VX=7$, and the side opposite is $XW=6$. 4. **Apply the formula:** $$\cos(\angle V) = \frac{5^2 + 7^2 - 6^2}{2 \times 5 \times 7} = \frac{25 + 49 - 36}{70} = \frac{38}{70}$$ 5. **Simplify the fraction:** $$\frac{38}{70} = \frac{\cancel{2} \times 19}{\cancel{2} \times 35} = \frac{19}{35}$$ 6. **Calculate the angle:** $$\angle V = \cos^{-1}\left(\frac{19}{35}\right) \approx \cos^{-1}(0.5429)$$ 7. **Use a calculator:** $$\angle V \approx 57.1^\circ$$ **Final answer:** $$m\angle V = 57.1^\circ$$