https://file.aiquickdraw.com/mathbot/file/1726756979979qp7zxsr3.jpg

Let's solve the problem step-by-step based on the information provided in the image.

We are given the following:

- \( \overrightarrow{BA} \) and \( \overrightarrow{BC} \) are opposite rays.
- \( \overrightarrow{BH} \) bisects \( \angle EBC \).
- \( \overrightarrow{BE} \) bisects \( \angle ABF \).

Also, we have these angle measures:
- \( m\angle ABF = (7b - 24)^\circ \)
- \( m\angle ABE = 2b^\circ \)

### Task:
We need to find \( m\angle EBF \).

### Steps to solve:

1. **Relationship between angles**:  
   Since \( \overrightarrow{BE} \) bisects \( \angle ABF \), we know that:
   \[
   m\angle ABE = m\angle EBF
   \]
   Therefore, \( m\angle EBF = 2b^\circ \).

2. **Expression for \( \angle ABF \)**:
   Since \( \overrightarrow{BE} \) bisects \( \angle ABF \), the full angle \( ABF \) is twice the measure of \( \angle ABE \):
   \[
   2 \times m\angle ABE = m\angle ABF
   \]
   Substituting the values:
   \[
   2 \times 2b = 7b - 24
   \]
   Simplifying this equation:
   \[
   4b = 7b - 24
   \]
   Solving for \( b \):
   \[
   4b - 7b = -24
   \]
   \[
   -3b = -24
   \]
   \[
   b = 8
   \]

3. **Finding \( m\angle EBF \)**:
   Now that we know \( b = 8 \), substitute this value into \( m\angle EBF = 2b \):
   \[
   m\angle EBF = 2 \times 8 = 16^\circ
   \]

Thus, the measure of \( \angle EBF \) is \( \boxed{16^\circ} \).

---

Would you like further details or have any questions? Here are five related questions to further your understanding:

1. How would the solution change if \( \overrightarrow{BE} \) did not bisect \( \angle ABF \)?
2. What geometric properties are used when bisectors of angles are involved?
3. Can you generalize the solution for different values of \( b \)?
4. What are the relationships between opposite rays and the angles they form?
5. How does the position of point \( E \) affect the angle bisectors?

**Tip:** Always double-check your algebra when solving for variables in geometric expressions!

In the figure, BA and BC are opposite rays. BH bisects ∠EBC and BE bisects ∠ABF. If m∠ABF = (7b - 24)° and m∠ABE = 2b°, find m∠EBF.

Learn how to solve a geometry problem involving angle bisectors and algebraic expressions. The problem asks to find m∠EBF, given m∠ABF = (7b - 24)° and m∠ABE = 2b°, and includes step-by-step solutions using the Angle Bisector Theorem and algebraic manipulation. Suitable for students in Grades 8-10.

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