Question: Solve for $x$ and $y$, then find $m\angle ABC$
Graph: two intersecting lines form four angles at point $B$; a diagonal line runs from top-left to bottom-right through $B$ with endpoint labeled $A$ above and below, and a horizontal line runs left-right through $B$ with endpoint labeled $C$ to the right. The $75^\circ$ angle is top-left of $B$, $(2x - 5)^\circ$ is top-right of $B$, and $y^\circ$ is bottom-left of $B$.
1. **State the problem:** We are given four angles around point $B$ formed by two intersecting lines and a diagonal line. The angles are $75^\circ$, $(2x - 5)^\circ$, $y^\circ$, and an unknown fourth angle. We need to solve for $x$ and $y$, then find the measure of $\angle ABC$.
2. **Identify relationships:** The angles around point $B$ form a full circle, so their measures add up to $360^\circ$.
3. **Use vertical angles:** Vertical angles are equal. The $75^\circ$ angle is vertical to the $y^\circ$ angle, so:
$$y = 75$$
4. **Use linear pairs:** Adjacent angles on a straight line sum to $180^\circ$. The angles $75^\circ$ and $(2x - 5)^\circ$ are adjacent on the diagonal line, so:
$$75 + (2x - 5) = 180$$
5. **Solve for $x$:**
$$75 + 2x - 5 = 180$$
$$70 + 2x = 180$$
$$2x = 180 - 70$$
$$2x = 110$$
$$x = \frac{110}{2}$$
$$x = 55$$
6. **Find $m\angle ABC$:** The angle $\angle ABC$ is the angle at $B$ between points $A$ and $C$. From the diagram, $\angle ABC$ corresponds to the angle labeled $(2x - 5)^\circ$. Substitute $x=55$:
$$m\angle ABC = 2(55) - 5 = 110 - 5 = 105^\circ$$
**Final answers:**
$$x = 55$$
$$y = 75$$
$$m\angle ABC = 105^\circ$$