Question: (b) Problem 15:
Find the measure of $\angle BEF$.
$(x + y)^\circ$
$96^\circ$
$y^\circ$
$2x^\circ$
$m\angle BEF = $
1. **State the problem:** We need to find the measure of $\angle BEF$ given the angles around points $B$ and $E$ with expressions involving $x$ and $y$.
2. **Analyze the angles at point $B$:** The angles around point $B$ are $96^\circ$ and $(x + y)^\circ$. Since these two angles are on a straight line, they are supplementary.
3. **Write the supplementary angle equation at $B$:**
$$
96 + (x + y) = 180
$$
4. **Simplify the equation:**
$$
96 + x + y = 180
$$
$$
x + y = 180 - 96 = 84
$$
5. **Analyze the angles at point $E$:** The angles $y^\circ$ and $2x^\circ$ are vertical angles formed by intersecting lines, so they are equal.
6. **Set the vertical angles equal:**
$$
y = 2x
$$
7. **Substitute $y = 2x$ into $x + y = 84$:**
$$
x + 2x = 84
$$
$$
3x = 84
$$
8. **Solve for $x$:**
$$
x = \frac{84}{3} = 28
$$
9. **Find $y$ using $y = 2x$:**
$$
y = 2 \times 28 = 56
$$
10. **Find $m\angle BEF$:** Given $m\angle BEF = y^\circ$, substitute $y$:
$$
m\angle BEF = 56^\circ
$$
**Final answer:**
$$
m\angle BEF = 56^\circ
$$