In the diagram, point $Q$ lies on $P R$ and point $S$ lies on $Q T$. What is the value of $x$ ?
(A) 10
(B) 30
(C) 50
(D) 40
(E) 20

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-020.jpg?height=274&width=461&top_left_y=1419&top_left_x=1255)
Since $Q S T$ is a straight angle, then $\angle Q S P=180^{\circ}-\angle T S P=180^{\circ}-50^{\circ}=130^{\circ}$.

Now $\angle R Q S$ is an exterior angle for $\triangle Q S P$.

This means that $\angle R Q S=\angle Q S P+\angle S P Q$.

Using the information that we know, $150^{\circ}=130^{\circ}+x^{\circ}$ and so $x=150-130=20$.

(Alternatively, we could have noted that $\angle R Q S$ and $\angle S Q P$ are supplementary and then used the sum of the angles in $\triangle Q S P$.)

ANSWER: (E)