1. **State the problem:** We are asked to find the measure of angle $\angle F$ in the circle with center $O$, given $m\angle DOE = 72^\circ$ and $m\angle EOG = 84^\circ$, and $EF$ is a diameter. 2. **Recall important facts:** - $EF$ is a diameter, so $\angle EGF$ is a right angle ($90^\circ$) because an angle inscribed in a semicircle is a right angle. - Central angles like $\angle DOE$ and $\angle EOG$ correspond to arcs on the circle. 3. **Find the arc measures:** - $m\widehat{DE} = m\angle DOE = 72^\circ$ - $m\widehat{EG} = m\angle EOG = 84^\circ$ 4. **Find the measure of arc $\widehat{DG}$:** $$m\widehat{DG} = m\widehat{DE} + m\widehat{EG} = 72^\circ + 84^\circ = 156^\circ$$ 5. **Find the measure of arc $\widehat{DF}$:** Since $EF$ is a diameter, the entire circle is $360^\circ$, and arc $\widehat{DF}$ is the complement of arc $\widehat{EG}$ plus arc $\widehat{DE}$, but more simply: $$m\widehat{DF} = 180^\circ$$ 6. **Find $m\angle F$:** $\angle F$ is an inscribed angle that intercepts arc $\widehat{DG}$. The measure of an inscribed angle is half the measure of its intercepted arc: $$m\angle F = \frac{1}{2} m\widehat{DG} = \frac{1}{2} \times 156^\circ = 78^\circ$$ **Final answer:** $$m\angle F = 78^\circ$$