a regular polygon has possible angles of rotational symmetry of 20°, 40°, and 80°. how many sides does the…

Explanation:

Step1: Recall the formula

The measure of the angle of rotational symmetry $\theta$ of a regular polygon with $n$ sides is given by $\theta=\frac{360^{\circ}}{n}$, where $n$ is a positive - integer. So, $n = \frac{360^{\circ}}{\theta}$. The number of sides $n$ must be an integer for all possible angles of rotational symmetry.

Step2: Check each angle

For $\theta = 20^{\circ}$, $n=\frac{360}{20}=18$. For $\theta = 40^{\circ}$, $n=\frac{360}{40}=9$. For $\theta = 80^{\circ}$, $n=\frac{360}{80}=4.5$ (not valid as $n$ must be an integer). Since the polygon has a valid angle of rotational symmetry of $20^{\circ}$ and the number of sides $n$ corresponding to $\theta = 20^{\circ}$ is 18.

Answer:

18