In the diagram, points $B, C$ and $D$ lie on a line. Also, $\angle A B C=90^{\circ}$ and $\angle A C D=150^{\circ}$. The value of $x$ is
(A) 30
(B) 45
(C) 90
(D) 150
(E) 60

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-003.jpg?height=215&width=414&top_left_y=261&top_left_x=1300)
Solution 1

Since $\angle B C D$ is a straight angle, then $\angle A C B=180^{\circ}-\angle A C D=180^{\circ}-150^{\circ}=30^{\circ}$.

Since the measures of the angles in $\triangle A B C$ add to $180^{\circ}$, then $x^{\circ}+90^{\circ}+30^{\circ}=180^{\circ}$ and so $x=180-120=60$.

## Solution 2

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

$$
\angle A C B=180^{\circ}-\angle A B C-\angle B A C=180^{\circ}-90^{\circ}-x^{\circ}=90^{\circ}-x^{\circ}
$$

Since $\angle B C D=180^{\circ}$, then

$$
\begin{aligned}
\angle A C B+\angle A C D & =180^{\circ} \\
\left(90^{\circ}-x^{\circ}\right)+150^{\circ} & =180^{\circ} \\
240-x & =180
\end{aligned}
$$

and so $x=60$.

ANSWER: (E)