To solve the problem, we need to identify the angle that forms both a linear pair and a vertical pair from the provided diagram.

Definitions:

• Linear Pair: Two angles that are adjacent and whose non-common sides form a straight line (they sum to 180 degrees).
• Vertical Pair: Two angles that are opposite each other when two lines intersect. Vertical angles are always congruent.

Step-by-Step Analysis:

1. Identify Linear Pairs:

   • In the diagram, angle $BFC$ forms a linear pair with $AFD$ because they are adjacent and lie on a straight line.
   • Similarly, angles like $EFA$ and $GFD$ form linear pairs since they are adjacent and form straight lines.

2. Identify Vertical Pairs:

   • Vertical angles are formed by the intersection of two lines. In this diagram, angle $BFC$ is opposite $EFD$, making them vertical angles.
   • Likewise, $CFG$ is vertical to $DFE$, $GFD$ is vertical to $AFB$, and so on.

3. Combine Both Conditions:

   • Angle $\angle BFC$ meets both conditions: it forms a linear pair with $AFD$ and is also part of a vertical pair with $EFD$.

Answer:

The correct answer is $\angle BFC$.

Would you like more details or explanations? Here are some related questions for further understanding:

1. What are other examples of linear pairs in this diagram?
2. Why are vertical angles always congruent?
3. How can we identify complementary or supplementary angles in a similar diagram?
4. What is the difference between adjacent and non-adjacent angles?
5. How does the concept of a transversal relate to this type of diagram?

Tip: When working with angles in intersecting lines, always look for linear pairs and vertical angles, as these help determine relationships between the angles efficiently.