The figure shows a regular ten-sided polygon with center $O$. What is the measure of angle $a$?

![](https://cdn.mathpix.com/cropped/2024_05_01_984603b876c0bc28095bg-13.jpg?height=334&width=340&top_left_y=827&top_left_x=961)
A) $15^{\circ}$
B) $18^{\circ}$
C) $20^{\circ}$
D) $30^{\circ}$
E) $36^{\circ}$

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Solution

## ALTERNATIVE B

The triangle $A O B$ is isosceles because the sides $O A$ and $O B$ are equal. Therefore, the angles $O A B$ and $O \hat{B} A$ are also equal, that is, both have a measure of $a$. We now notice that the central angle $A \hat{O} B$ measures $\frac{4}{10} \times 360^{\circ}=144^{\circ}$. Since the sum of the interior angles of a triangle is $180^{\circ}$, it follows that $2 a+144^{\circ}=180^{\circ}$. Therefore, $a=\frac{180^{\circ}-144^{\circ}}{2}=\frac{36^{\circ}}{2}=18^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_984603b876c0bc28095bg-38.jpg?height=374&width=414&top_left_y=561&top_left_x=730)