In the diagram, $A B$ and $C D$ intersect at $E$. If $\triangle B C E$ is equilateral and $\triangle A D E$ is a right-angled triangle, what is the value of $x$ ?
(A) 90
(B) 60
(C) 25
(D) 45
(E) 30

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-013.jpg?height=268&width=406&top_left_y=1232&top_left_x=1315)
Since $\triangle B C E$ is equilateral, then each of its three interior angles measures $60^{\circ}$.

Vertically opposite angles are equal in measure, and so $\angle D E A=\angle B E C=60^{\circ}$. In $\triangle A D E$, the sum of the three interior angles is $180^{\circ}$.

Thus, $x^{\circ}+90^{\circ}+60^{\circ}=180^{\circ}$ or $x+150=180$ and so $x=30$.

ANSWER: (E)