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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.

In the diagram, which angle is part of a linear pair and part of a vertical pair?

This problem asks students to identify an angle that forms both a linear pair and a vertical pair from a given diagram. The solution explores the relationships between linear pairs and vertical angles, providing a detailed step-by-step approach. Suitable for students in Grades 7-9.

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