https://file.aiquickdraw.com/mathbot/file/1727032640887qtjjk9z3.jpg

The problem asks to apply the **vertical angles theorem** to solve for the measure of \( \angle BFA \) in the diagram.

From the image, we see that:
- \( \angle BFA = (4x - 12)^\circ \)
- \( \angle CFB = (2x + 18)^\circ \)

The **vertical angles theorem** states that vertical angles are congruent. Therefore, we can set these two expressions equal to each other:

\[
(4x - 12)^\circ = (2x + 18)^\circ
\]

### Solving for \( x \):

1. Subtract \( 2x \) from both sides:
\[
4x - 12 - 2x = 2x + 18 - 2x
\]
\[
2x - 12 = 18
\]

2. Add 12 to both sides:
\[
2x - 12 + 12 = 18 + 12
\]
\[
2x = 30
\]

3. Divide both sides by 2:
\[
x = 15
\]

### Now substitute \( x = 15 \) back into the expression for \( \angle BFA \):

\[
\angle BFA = 4x - 12 = 4(15) - 12 = 60 - 12 = 48^\circ
\]

So, the measure of \( \angle BFA \) is **48°**, which corresponds to **Option C**.

---

Would you like more details or have any questions? 

Here are 5 related questions to expand on this problem:
1. How can you prove the vertical angles theorem?
2. What is the relationship between adjacent angles and vertical angles?
3. How would the solution change if the angles were supplementary instead of congruent?
4. Can vertical angles ever be complementary? Why or why not?
5. What are other theorems involving angle relationships in geometry?

**Tip:** Always look for vertical, supplementary, and complementary angles when analyzing geometric diagrams with intersecting lines.

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

Learn how to apply the vertical angles theorem to solve for the measure of ∠BFA in a diagram with intersecting lines. This solution uses algebra to solve for x and determine the angle's measure. Suitable for students in Grades 8-10.

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