1. **State the problem:** Given $m\angle N\hat{O} = 238^\circ$, find $m\angle P\hat{O}Q$ and $m\angle P\hat{Q}R$ based on the circle and points described. 2. **Recall circle angle properties:** - The angle $m\angle N\hat{O}$ is a central angle measuring 238°. - Central angles measure the arc they intercept. - Angles around a point sum to 360°. - Inscribed angles subtending the same arc are half the measure of the central angle. 3. **Find $m\angle P\hat{O}Q$:** - Since $m\angle N\hat{O} = 238^\circ$, the remaining angle around point $O$ is $360^\circ - 238^\circ = 122^\circ$. - $m\angle P\hat{O}Q$ is the angle adjacent to $m\angle N\hat{O}$, so $m\angle P\hat{O}Q = 122^\circ$. 4. **Find $m\angle P\hat{Q}R$:** - $\angle P\hat{Q}R$ is an inscribed angle subtending the same arc as $\angle N\hat{O}$. - Inscribed angle measure is half the measure of the intercepted arc. - So, $m\angle P\hat{Q}R = \frac{1}{2} \times 238^\circ = 119^\circ$. **Final answers:** $$m\angle P\hat{O}Q = 122^\circ$$ $$m\angle P\hat{Q}R = 119^\circ$$