To solve for the measure of angle $\angle DFC$, we need to carefully analyze the geometry based on the given information.

Steps to solve:

1. Given Information:

   • Points A, F, D lie on one line.
   • Points B, F, E lie on another line.
   • The lines intersect at point F.
   • $\angle AFE = 125^\circ$ (Angle between AF and FE).
   • $\angle BFC = 45^\circ$ (Angle between BF and FC).

2. Understand the Relationships:

   • The angles $\angle AFE$ and $\angle BFC$ are formed by the intersection of the two lines at point F.
   • These angles are related to each other by the geometry of the intersecting lines and the extended line from F to C.

3. Analyze $\angle DFC$:

   • $\angle DFC$ is formed between the line segment $DF$ and the line segment $FC$.
   • Since $\angle BFC = 45^\circ$, and considering the angle relationship around point F, we can deduce that $\angle DFC$ is supplementary to $\angle BFC$.

   $\angle DFC + \angle BFC = 180^\circ$ Substituting the known value for $\angle BFC$:

   $\angle DFC + 45^\circ = 180^\circ$

   Solving for $\angle DFC$:

   $\angle DFC = 180^\circ - 45^\circ = 135^\circ$

Final Answer: $\angle DFC = 135^\circ$.

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Would you like further clarification on any steps or concepts? Here are some related questions to explore:

1. How does the concept of supplementary angles apply in this problem?
2. What is the relationship between vertical angles and the angle pairs here?
3. How can the angles around a point be used to solve other geometric problems?
4. Why is the sum of angles on a straight line always 180°?
5. Can this method be applied to any intersection of two lines, regardless of the angles given?

Tip: When dealing with intersecting lines, always check for supplementary or vertically opposite angle relationships. They can simplify the problem!