1. **State the problem:** We need to find the measure of angle $F$ given two expressions for angles related to points on a circle: $2x - 85^\circ$ and $x - 22^\circ$. 2. **Understand the relationship:** Since points $G, D, F, E$ lie on the circle and triangles $GDF$ and $FDE$ are inscribed, angles subtended by the same arc or related arcs have special relationships. Here, the two given angles are likely related by the property that the sum of angles around point $F$ on the circle equals $180^\circ$ (angles in a triangle sum to $180^\circ$). 3. **Set up the equation:** Assuming the two angles are adjacent and supplementary, $$ (2x - 85) + (x - 22) = 180 $$ 4. **Simplify the equation:** $$ 2x - 85 + x - 22 = 180 $$ $$ 3x - 107 = 180 $$ 5. **Solve for $x$:** $$ 3x = 180 + 107 $$ $$ 3x = 287 $$ $$ x = \frac{287}{3} $$ 6. **Calculate $x$:** $$ x = 95.666\ldots $$ 7. **Find $m \angle F$:** Given $m \angle F = x - 22$, substitute $x$: $$ m \angle F = 95.666\ldots - 22 = 73.666\ldots $$ 8. **Final answer:** $$ m \angle F \approx 73.67^\circ $$ This is the measure of angle $F$.