In the diagram, the value of $x$ is
(A) 40
(B) 45
(C) 50
(D) 55
(E) 60

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-081.jpg?height=454&width=263&top_left_y=1613&top_left_x=1477)
From the diagram, $B E=A D$ and $A E=C D$, so

$A C=A D+C D=B E+A E=A B$ so triangle $A B C$ is isosceles.

Therefore, $\angle A C B=\angle A B C=80^{\circ}$ and so

$\angle B A C=180^{\circ}-\angle A B C-\angle A C B=180^{\circ}-80^{\circ}-80^{\circ}=20^{\circ}$.

Considering triangle $A E D$ next,

$\angle A E D=180^{\circ}-\angle A D E-\angle E A D=180^{\circ}-30^{\circ}-20^{\circ}=130^{\circ}$.

But $x^{\circ}=180^{\circ}-\angle A E D=180^{\circ}-130^{\circ}=50^{\circ}$ so $x=50$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-360.jpg?height=344&width=201&top_left_y=1210&top_left_x=1512)