The problem involves a quadrilateral $DGFE$, and you're asked to determine the measure of $\angle DFG$ given that $\angle DFE = 34^\circ$.

Step-by-step solution:

1. Symmetry observation: The diagram suggests symmetry. Given that the angles at $G$ and $E$ seem to mirror each other, we can assume that $\angle DFE$ and $\angle DFG$ are related geometrically.

2. Isosceles triangle (based on symmetry): Since $\triangle DFG$ appears isosceles (because the legs $DG$ and $DF$ seem equal in length based on symmetry), $\angle DFG$ will be half of the remaining angle not covered by $\angle DFE$.

3. Triangle angle sum: In $\triangle DFG$, the sum of angles must be $180^\circ$. The total angles at $F$ and $E$ should equal $180^\circ - 34^\circ = 146^\circ$.

4. Conclusion: As $\triangle DFG$ appears isosceles, $\angle DFG = 17^\circ$ (half of the remaining angle of $146^\circ$).

Thus, the correct answer is: $\boxed{17^\circ}$

Answer: $d.$ 17°

Do you need further clarification or have any questions?

Here are 5 related questions to explore further:

1. What would happen if $\angle DFE$ was 40°? How would $\angle DFG$ change?
2. Could this problem be solved using trigonometry?
3. Is it possible that $\triangle DFG$ is not isosceles? How would that affect the solution?
4. How do we find the interior angle sum of a quadrilateral like $DGFE$?
5. How does symmetry help in solving geometric problems?

Tip: In geometry, identifying symmetry or congruent parts of a figure can often simplify complex problems significantly!