1. **Problem Statement:** Calculate the size of angle $\hat{O}_1$ given that $O$ is the center of circle $HEATR$, $AOF$ is parallel to $EH$, $\hat{F}_2 = 60^\circ$, and $\hat{R}_1 = 28^\circ$. 2. **Key Information and Rules:** - $O$ is the center of the circle, so $OA$, $OE$, etc. are radii. - $AOF$ is parallel to $EH$, so corresponding angles are equal. - Angles subtended by the same chord at the center and circumference have specific relationships. 3. **Step-by-step Solution:** (i) To find $\hat{O}_1$: - Since $AOF$ is parallel to $EH$, angle $\hat{O}_1$ corresponds to angle $\hat{F}_2$ by alternate interior angles. - Given $\hat{F}_2 = 60^\circ$, therefore $\hat{O}_1 = 60^\circ$. 4. **Reasoning:** - Parallel lines imply alternate interior angles are equal. - Hence, $\hat{O}_1 = \hat{F}_2 = 60^\circ$. **Final answer:** $$\hat{O}_1 = 60^\circ$$