1. **State the problem:** We are given a circle with center $A$ and points $W, X, Y, Z$ on the circle. $\overline{YW}$ and $\overline{XZ}$ are diameters, which means each subtends a $180^\circ$ arc. We want to find the measure of the major arc $\stackrel{\large{\frown}}{YWZ}$ in degrees. 2. **Interpret the given angles:** - $\angle X A Y = 11N$ degrees. - $\angle Y A Z = 5N + 4$ degrees. Since $W, X, Y, Z$ lie on the circle, and $WA$ and $YA$ are radii, these central angles correspond to arcs between those points. 3. **Use the fact that the points are clockwise:** The order around the circle is $W \to X \to Y \to Z$, and using the central angles we can find the arc measures. 4. The diameter $\overline{YW}$ divides the circle into two semicircles of $180^\circ$ each. Therefore: - Arc $YW = 180^\circ$ - Arc $W Y$ opposite would also be $180^\circ$ 5. Similarly, diameter $\overline{XZ}$ divides the circle into arcs $X Z$ and $Z X$, each $180^\circ$. 6. **Identify arcs corresponding to angles at center:** - $\angle X A Y = 11N$ corresponds to arc $XY$. - $\angle Y A Z = 5N+4$ corresponds to arc $YZ$. 7. Total arc from $X$ to $Z$ going through $Y$ is arc $XY + YZ$: $$\text{arc } XZ = 11N + 5N + 4 = 16N + 4$$ But from step 5 we know $\text{arc } XZ = 180^\circ$ because $\overline{XZ}$ is a diameter. 8. Set up the equation: $$16N + 4 = 180$$ $$16N = 176$$ $$N = 11$$ 9. Calculate angle values: - $\angle X A Y = 11 \times 11 = 121^\circ$ - $\angle Y A Z = 5 \times 11 + 4 = 55 + 4 = 59^\circ$ 10. We want the measure of the major arc $\stackrel{\large{\frown}}{YWZ}$. Note this arc goes from $Y$ to $W$ to $Z$ around the circle clockwise. 11. Arc $YW = 180^\circ$ (diameter), arc $YZ = 59^\circ$, so arc $YWZ$ = arc $YW +$ arc $WZ$. 12. Since $W, X, Y, Z$ are clockwise, and we know arc $XZ = 180^\circ$, arc $WZ$ is the arc complement to $X Z$ (since $W$ and $X$ are endpoints of $W X$ diameter). The full circle is $360^\circ$. 13. To find arc $WZ$: - $\text{arc } WZ = 360^\circ - \text{arc } ZW$. - Given arc $YW = 180^\circ$, then arc $WZ$ = arc $XZ$ (since diameters $YW$ and $XZ$ are perpendicular to each other center-wise). Actually, from the order $W, X, Y, Z$, moving clockwise: - Arc $YW$ = $180^\circ$ (diameter) - Arc $YZ$ = $59^\circ$ - Arc $WY$ and $WZ$ relationship imply major arc $YWZ = 180^\circ +$ arc $YZ = 180^\circ + 59^\circ = 239^\circ$ 14. **Final answer:** The arc measure of the major arc $\stackrel{\large{\frown}}{YWZ}$ is $$239^\circ$$