A hexagon is called equiangular when it has six equal internal angles. Consider the equiangular hexagon $A B C D E F$ with sides $3, y, 5, 4, 1$, and $x$, as shown in the figure below. Determine the unknown lengths $x$ and $y$.

![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-35.jpg?height=446&width=462&top_left_y=925&top_left_x=891)
Solution

Since a hexagon can be divided into 4 triangles through its diagonals, the sum of its interior angles is $180^{\circ}(6-2)=720^{\circ}$. Given that it is equiangular, each of the interior angles will measure $\frac{720^{\circ}}{6}=120^{\circ}$. Knowing this, by extending the sides, we will form, as indicated below, smaller equilateral triangles externally on three of its sides and a larger equilateral triangle $\triangle X Y Z$ that will contain it.

![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-35.jpg?height=523&width=576&top_left_y=1897&top_left_x=840)

Since the sides of the triangle $\triangle X Y Z$ are equal, we have

$$
3+y+5=5+4+1=1+x+3
$$

Therefore, $x=6$ and $y=2$.