Archaeologists found a gold necklace made of plates in the shape of regular pentagons. Each of these plates is connected to two other plates, as illustrated in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_01_64d7a736a938f072abd8g-1.jpg?height=332&width=600&top_left_y=1002&top_left_x=565)

Figure 51.1

How many plates form the necklace?
The internal angle of a regular pentagon measures $108^{\circ}$. Therefore, the internal angle of the polygon determined by the necklace measures $360^{\circ}-108^{\circ}-108^{\circ}=144^{\circ}$. We must then find $n$ such that

$$
\frac{180^{\circ}(n-2)}{n}=144^{\circ}
$$

Solving this equation, we get $n=10$. Therefore, ten plates form the necklace.