1. **State the problem:** Given a circle with points L, X, W, and N on its circumference, and two intersecting lines crossing outside the circle at point V, where $m\angle WX = 62^\circ$ and $m\angle NL = 176^\circ$, find $m\angle V$. 2. **Recall the rule for angles formed by two secants intersecting outside a circle:** $$m\angle V = \frac{1}{2} |m\overset{\frown}{WL} - m\overset{\frown}{XN}|$$ where $m\overset{\frown}{WL}$ and $m\overset{\frown}{XN}$ are the measures of the intercepted arcs. 3. **Identify the intercepted arcs:** - The angle at $V$ formed by secants $VLX$ and $VNW$ intercepts arcs $WL$ and $XN$. 4. **Use the given angles to find the arcs:** - $m\angle WX = 62^\circ$ is an inscribed angle intercepting arc $WL$, so: $$m\angle WX = \frac{1}{2} m\overset{\frown}{WL} \implies m\overset{\frown}{WL} = 2 \times 62^\circ = 124^\circ$$ - $m\angle NL = 176^\circ$ is an inscribed angle intercepting arc $XN$, so: $$m\angle NL = \frac{1}{2} m\overset{\frown}{XN} \implies m\overset{\frown}{XN} = 2 \times 176^\circ = 352^\circ$$ 5. **Calculate $m\angle V$ using the formula:** $$m\angle V = \frac{1}{2} |m\overset{\frown}{WL} - m\overset{\frown}{XN}| = \frac{1}{2} |124^\circ - 352^\circ| = \frac{1}{2} | -228^\circ| = \frac{1}{2} \times 228^\circ = 114^\circ$$ 6. **Final answer:** $$m\angle V = 114^\circ$$