Question

if a regular hexagon is rotated about its center, which angles would result in the hexagon being mapped back onto itself? (there are multiple correct answers.) 90° 60° 120° 45° 180°

Explanation:

Step1: Recall rotational symmetry of hexagon

A regular hexagon has rotational symmetry. The central angle of a regular hexagon is calculated as $\frac{360^{\circ}}{n}$, where $n = 6$ (number of sides). So the central - angle is $\frac{360^{\circ}}{6}=60^{\circ}$.

Step2: Determine rotation angles

The angles of rotation that map a regular hexagon onto itself are multiples of the central - angle. So the angles are $60^{\circ},120^{\circ},180^{\circ}$ (since $120^{\circ}=2\times60^{\circ}$ and $180^{\circ}=3\times60^{\circ}$). A $90^{\circ}$ rotation and a $45^{\circ}$ rotation will not map a regular hexagon onto itself.

Answer:

$60^{\circ},120^{\circ},180^{\circ}$