The problem involves finding the measure of $\angle DFG$ given that $\overline{DE} \parallel \overline{GF}$ and $\angle DFE = 125^\circ$.

Solution:

1. Since $\overline{DE} \parallel \overline{GF}$ and the given angles involve a transversal line intersecting the parallel lines, we use the concept of alternate interior angles and supplementary angles.

2. $\angle DFE = 125^\circ$ is supplementary to its adjacent angle at $\angle DFG$ because they are on a straight line.

3. Using the property of supplementary angles: $\angle DFG + \angle DFE = 180^\circ$ Substituting $\angle DFE = 125^\circ$: $\angle DFG = 180^\circ - 125^\circ = 55^\circ$

Final Answer:

$m \angle DFG = 55^\circ$

Would you like a detailed explanation of the reasoning, or do you have any further questions?

Related Questions:

1. How are supplementary angles related to parallel lines and a transversal?
2. What are alternate interior angles, and how are they used in geometry problems?
3. What is the sum of angles around a point, and why is it important here?
4. Could we solve this problem using a different method? How?
5. How can this concept be applied to other geometric problems involving parallel lines?

Tip:

Remember, when two lines are parallel and cut by a transversal, alternate interior angles are always congruent, and consecutive interior angles are supplementary!