## Task 3 - 170823

The figure shows a five-pointed star, whose tips $A, B, C$, $D, E$ are the vertices of a regular pentagon.

Determine the size of the angle $\angle A D B$!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0694.jpg?height=366&width=374&top_left_y=314&top_left_x=1412)
Let $A, B, C, D, E$ be the vertices of a regular pentagon, all lying on the periphery of a circle; let the center of this circle be $M$.

Connecting $M$ to the aforementioned vertices results in 5 congruent triangles $A M B, B M C$, $C M D, D M E, E M A$. The sum of their angles with vertex $M$ forms a full angle, so each of these angles is $360^{\circ}: 5=72^{\circ}$.

Since in a circle every peripheral angle is half as large as the central angle subtending the same arc, we have

$$
\angle A D B=\frac{1}{2} \angle A M B=\frac{1}{2} \cdot 72^{\circ}=36^{\circ}
$$