In the diagram, $P W$ is parallel to $Q X, S$ and $T$ lie on $Q X$, and $U$ and $V$ are the points of intersection of $P W$ with $S R$ and $T R$, respectively. If $\angle S U V=120^{\circ}$ and $\angle V T X=112^{\circ}$, what is the measure of $\angle U R V$ ?
(A) $52^{\circ}$
(B) $56^{\circ}$
(C) $60^{\circ}$
(D) $64^{\circ}$
(E) $68^{\circ}$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-082.jpg?height=358&width=445&top_left_y=263&top_left_x=1271)
Solution 1

Since $S U R$ is a straight line, then $\angle R U V=180^{\circ}-\angle S U V=180^{\circ}-120^{\circ}=60^{\circ}$.

Since $P W$ and $Q X$ are parallel, then $\angle R V W=\angle V T X=112^{\circ}$.

Since $U V W$ is a straight line, then $\angle R V U=180^{\circ}-\angle R V W=180^{\circ}-112^{\circ}=68^{\circ}$.

Since the measures of the angles in a triangle add to $180^{\circ}$, then

$$
\angle U R V=180^{\circ}-\angle R U V-\angle R V U=180^{\circ}-60^{\circ}-68^{\circ}=52^{\circ}
$$

## Solution 2

Since $S U R$ is a straight line, then $\angle R U V=180^{\circ}-\angle S U V=180^{\circ}-120^{\circ}=60^{\circ}$.

Since $P W$ and $Q X$ are parallel, then $\angle R S T=\angle R U V=60^{\circ}$.

Since $S T X$ is a straight line, then $\angle R T S=180^{\circ}-\angle V T X=180^{\circ}-112^{\circ}=68^{\circ}$.

Since the measures of the angles in a triangle add to $180^{\circ}$, then

$$
\angle U R V=\angle S R T=180^{\circ}-\angle R S T-\angle R T S=180^{\circ}-60^{\circ}-68^{\circ}=52^{\circ}
$$

ANSWER: (A)