https://file.aiquickdraw.com/mathbot/file/1726767593906i2fo5u5d.jpg

To solve for the measure of ∠BFA using the vertical angles theorem, we can follow these steps:

1. **Identify Vertical Angles:**
   - The vertical angles theorem states that vertical angles are congruent (equal). In the diagram, ∠BFA and ∠CFD are vertical angles, so:
     \[
     m∠BFA = m∠CFD
     \]
   - According to the diagram:
     \[
     m∠BFA = (4x - 12)^\circ
     \]
     and
     \[
     m∠CFD = (2x + 18)^\circ
     \]

2. **Set the angles equal to each other:**
   Since ∠BFA and ∠CFD are vertical angles:
   \[
   4x - 12 = 2x + 18
   \]

3. **Solve for \( x \):**
   \[
   4x - 2x = 18 + 12
   \]
   \[
   2x = 30
   \]
   \[
   x = 15
   \]

4. **Substitute \( x = 15 \) back into the expression for ∠BFA:**
   \[
   m∠BFA = 4x - 12 = 4(15) - 12 = 60 - 12 = 48^\circ
   \]

Thus, the measure of ∠BFA is **48°**, which corresponds to option C.

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Would you like more details on any step?

Here are five questions you could explore further:
1. How does the vertical angles theorem work in other geometric scenarios?
2. What are supplementary angles, and how would they apply in this diagram?
3. Can you solve for any other angles in the diagram based on the value of \( x \)?
4. What are some other geometric theorems used to solve for unknown angles?
5. How do alternate interior angles relate to parallel lines and transversals?

**Tip:** Always double-check if the given angles are vertical or supplementary, as this affects the equation setup.

Apply the vertical angles theorem to solve for the m∠BFA in the diagram.

Learn how to solve for m∠BFA using the vertical angles theorem. Step-by-step breakdowns of the angles and algebra involved to find that m∠BFA = 48°. Suitable for Grades 8-10 students.

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