In the figure, the two triangles $\triangle A B C$ and $\triangle D E F$ are equilateral. What is the value of the angle $x$?

(a) $30^{\circ}$

(b) $40^{\circ}$

(c) $50^{\circ}$

(d) $60^{\circ}$

(e) $70^{\circ}$

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-009.jpg?height=394&width=637&top_left_y=780&top_left_x=732)
The correct option is (b).

Since $\triangle A B C$ and $\triangle D E F$ are equilateral triangles, each of their internal angles measures $60^{\circ}$. In the triangle $\triangle A G D$ we have

$$
G \widehat{A} D=180^{\circ}-75^{\circ}-60^{\circ}=45^{\circ} \text { and } \mathrm{G} \widehat{D A}=180^{\circ}-65^{\circ}-60^{\circ}=55^{\circ}
$$

Therefore, $A \widehat{G} D=180^{\circ}-45^{\circ}-55^{\circ}=80^{\circ}$. Thus, in the triangle $\triangle C G H$, we have

$$
x+80^{\circ}+60^{\circ}=180^{\circ}, \text { hence } x=40^{\circ} \text {. }
$$