A8. The vertices of a regular pentadecagon are connected as shown in the picture. (Beware: the dimensions in the picture are not exact!) What is the angle, indicated by an arc, between $A C$ and $B D$?
A) $130^{\circ}$
B) $132^{\circ}$
C) $135^{\circ}$
D) $136^{\circ}$
E) $137.5^{\circ}$

## B-questions

![](https://cdn.mathpix.com/cropped/2024_04_17_a91c336121b9423a3a1eg-2.jpg?height=426&width=420&top_left_y=1095&top_left_x=1503)
A8. B) $132^{\circ}$ Let the center of the pentadecagon be $M$ and the intersection of $A C$ and $B D$ be point $S$ (see the figure). In quadrilateral $M R S T$, we see that $\angle M R S=90^{\circ}$ and $\angle S T M=90^{\circ}$. Furthermore, we see that $\angle T M R=\frac{2}{15} \cdot 360^{\circ}=48^{\circ}$. The sum of the angles in a quadrilateral is $360^{\circ}$, so we have: $\angle R S T=360^{\circ}-2 \cdot 90^{\circ}-48^{\circ}=132^{\circ}$. Note that $\angle R S T$ and $\angle B S C$ are vertical angles. The required angle is thus also $132^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_7effb8aeb7220f8b3452g-2.jpg?height=457&width=485&top_left_y=771&top_left_x=1385)