Question

A regular polygon has possible angles of rotational symmetry of 20°, 40°, and 80°. How many sides does the polygon have?

Answer 1

Angle of rotational symmetry is the angle one can turn a figure and the figure appears unchanged. (learner.org)

Also, any multiple of the least angle of rotational symmetry is also an angle of rotational symmetry.  Thus, 40 and 80 are automatically angles of rotational symmetry if 20 is one.

So the regular polygon must have at least 360/20=18 sides.
note: 36, 54,... sides will also work.  The questions appears to mean to ask for the minimum number of sides of the polygon.

Answer 2

Answer:  The polygon has 18 sides.

Step-by-step explanation:  Given that a regular polygon has possible angles of rotational symmetry of 20°, 40°, and 80°.

We are to find the number of sides of the polygon.

Let α be the smallest angle of rotational symmetry of the regular polygon.

Then,

$\alpha=gcd(20^\circ,40^\circ,80^\circ)=20^\circ.$

Therefore, the total number of sides of the polygon will be

$n=\dfrac{360^\circ}{20^\circ}=18.$

Thus, the number of sides of the polygon is 18.