Task A-1.5. (8 points)

In the figure, there is a regular hexagon $A B C D E F$ and a regular pentagon $E F G H I$.

Determine the angle $\varangle F A G$.

![](https://cdn.mathpix.com/cropped/2024_05_30_59925bf190fc107f19b7g-03.jpg?height=437&width=554&top_left_y=455&top_left_x=268)
## Solution.

Let $S$ be the center of a regular hexagon, and $T$ be the center of a regular pentagon.

![](https://cdn.mathpix.com/cropped/2024_05_30_59925bf190fc107f19b7g-03.jpg?height=534&width=688&top_left_y=1201&top_left_x=273)

We have $\varangle A S F = 360^{\circ} / 6 = 60^{\circ}$, so $\varangle A F E = 120^{\circ}$. Also, $\varangle G T F = 360^{\circ} / 5 = 72^{\circ}$, so $\varangle E F G = 108^{\circ}$. Therefore, $\varangle A F G = 360^{\circ} - \varangle A F E - \varangle E F G = 132^{\circ}$.

Triangle $A F G$ is isosceles, with $|A F| = |F G|$, so $\varangle F A G = (180^{\circ} - \varangle A F G) / 2 = 24^{\circ}$.

For 4 points: at least "angles $120^{\circ}, 108^{\circ}$ and triangle $A F G$ is isosceles" or "angle $\varangle A F G$ is $132^{\circ}$" or a good method with a calculation error.

For 8 points: the correct result and method.