5. At point $C$, an equilateral triangle $A C G$, a regular pentagon $C D E F G$, and a regular octagon BHIJKLDC meet (see figure). Determine the angles of triangle $A B C$.

![](https://cdn.mathpix.com/cropped/2024_06_07_5a4348e8f3c4edef190ag-02.jpg?height=468&width=628&top_left_y=1442&top_left_x=1185)

Solve the problems independently. You have 150 minutes for solving.

The use of notes, literature, or a pocket calculator is not allowed.

Mathematical Competition for High School Students in Slovenia

Optional Competition

March 30, 2005

## Problems for 2nd Year
I/5. Let's recall that the internal angle of a regular $n$-sided polygon measures $\frac{n-2}{n} 180^{\circ}$. The internal angle of an equilateral triangle measures $60^{\circ}$, of a regular pentagon $\frac{3}{5} 180^{\circ}=108^{\circ}$, and of a regular octagon $\frac{6}{8} 180^{\circ}=135^{\circ}$. Triangle $A B C$ is isosceles with the angle at vertex $C$ being $360^{\circ}-\left(60^{\circ}+108^{\circ}+135^{\circ}\right)=57^{\circ}$ and the angles at the base $A B$ being $\frac{1}{2}\left(180^{\circ}-57^{\circ}\right)=61.5^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_5a4348e8f3c4edef190ag-07.jpg?height=60&width=1694&top_left_y=667&top_left_x=181)

Calculation of the internal angle of the pentagon ....................................................................

Calculation of the internal angle of the octagon .................................................................................

Calculation of the angle at the vertex of the isosceles triangle $A B C$ ...................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_5a4348e8f3c4edef190ag-07.jpg?height=60&width=1694&top_left_y=867&top_left_x=181)