1. **State the problem:** We need to find the measure of angle $m\angle DFE$ given a portion of a circle with chord $D$ and points $D$, $F$, and $E$ on or near the circle. 2. **Recall the circle angle rules:** - An inscribed angle in a circle is half the measure of its intercepted arc. - If $\angle DFE$ is an inscribed angle intercepting arc $DE$, then $$m\angle DFE = \frac{1}{2} m\overset{\frown}{DE}$$ 3. **Analyze the given options:** - The options are $130^\circ$, $50^\circ$, $65^\circ$, and $230^\circ$. 4. **Determine the intercepted arc:** - Since $m\angle DFE$ is an inscribed angle, the intercepted arc must be twice the angle measure. 5. **Check which option fits a valid arc measure:** - If $m\angle DFE = 65^\circ$, then the intercepted arc is $2 \times 65^\circ = 130^\circ$. - $130^\circ$ is a reasonable arc measure for a circle. 6. **Conclusion:** - Therefore, $m\angle DFE = 65^\circ$. **Final answer:** $$m\angle DFE = 65^\circ$$