The problem involves bisected angles, and we are given expressions for some of the angles.

Key Information:

• $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are opposite rays, meaning $\angle ABC$ forms a straight line (180°).
• $\overrightarrow{BH}$ bisects $\angle EBC$, and $\overrightarrow{BE}$ bisects $\angle ABF$.
• The measures of the angles are:
  • $m\angle ABF = (7b - 24)^\circ$
  • $m\angle ABE = 2b^\circ$

We are tasked with finding $m\angle EBF$, which is half of $\angle ABF$, because $\overrightarrow{BE}$ bisects it.

Steps:

1. Determine $m\angle ABF$: $m\angle ABF = (7b - 24)^\circ$

2. Find $m\angle ABE$: We are given $m\angle ABE = 2b^\circ$. Since $\overrightarrow{BE}$ bisects $\angle ABF$, it divides the angle into two equal parts: $m\angle ABE + m\angle EBF = m\angle ABF$

3. Set up the equation: Using the fact that $m\angle ABE = m\angle EBF$: $2 \cdot m\angle ABE = m\angle ABF$ Substituting the known values: $2 \cdot (2b) = 7b - 24$ Simplifying: $4b = 7b - 24$

4. Solve for $b$: $7b - 4b = 24$ $3b = 24$ $b = 8$

5. Calculate $m\angle ABF$: Substituting $b = 8$ into the expression for $m\angle ABF$: $m\angle ABF = 7(8) - 24 = 56 - 24 = 32^\circ$

6. Find $m\angle EBF$: Since $m\angle ABE = m\angle EBF$ and $m\angle ABE = 2b$, we substitute $b = 8$: $m\angle EBF = 2(8) = 16^\circ$

Final Answer:

$m\angle EBF = 16^\circ$.

Would you like more details or have any further questions? Here are some related questions:

1. How do you determine if two rays are opposite?
2. What does it mean for a ray to bisect an angle?
3. How can you find the measure of angles in bisected geometric figures?
4. What properties define angles formed by opposite rays?
5. How can this technique be applied to other geometric bisectors?

Tip: When solving angle bisector problems, setting up an equation with known relationships can simplify finding unknowns.