In the diagram, $Q$ is on $P R$. The value of $x$ is
(A) 50
(B) 80
(C) 100
(D) 16.7
(E) 130

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-044.jpg?height=309&width=448&top_left_y=1770&top_left_x=1245)
Solution 1

Since $\angle P Q S$ is an exterior angle of $\triangle Q S R$, then $\angle P Q S=\angle Q S R+\angle S R Q$.

Thus, $2 x^{\circ}=x^{\circ}+50^{\circ}$ and so $2 x=x+50$ or $x=50$.

## Solution 2

Since $\angle P Q S$ and $\angle S Q R$ form a straight angle, then $\angle P Q S$ and $\angle S Q R$ are supplementary.

Therefore, $\angle S Q R=180^{\circ}-\angle P Q S=180^{\circ}-2 x^{\circ}$.

Since the sum of the angles in $\triangle S Q R$ is $180^{\circ}$, then

$$
\begin{aligned}
\angle S Q R+\angle Q S R+\angle Q R S & =180^{\circ} \\
\left(180^{\circ}-2 x^{\circ}\right)+x^{\circ}+50^{\circ} & =180^{\circ} \\
50^{\circ}-x^{\circ} & =0^{\circ} \\
x^{\circ} & =50^{\circ}
\end{aligned}
$$

Therefore, $x=50$.

ANSwER: (A)