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!