1. **Stating the problem:** We need to find the measure of angle $\angle G$ formed by points $G$, $F$, and $E$ on a circle. 2. **Given information:** The measure of the arc $\overset{\frown}{GF}$ is $104^\circ$. 3. **Key rule:** The measure of an angle formed by two chords intersecting on the circumference of a circle is half the measure of the intercepted arc. 4. **Applying the rule:** Since $\angle G$ intercepts arc $\overset{\frown}{GF}$, we have $$m\angle G = \frac{1}{2} m\overset{\frown}{GF}$$ 5. **Substitute the given arc measure:** $$m\angle G = \frac{1}{2} \times 104^\circ$$ 6. **Calculate:** $$m\angle G = 52^\circ$$ 7. **Answer:** The measure of angle $\angle G$ is $52^\circ$.