the regular polygon below is to be rotated about its center. which angle of rotation would carry the figure…

Explanation:

Step1: Recall rotation - symmetry formula

For a regular polygon with (n) sides, the angle of rotation (\theta) that maps the polygon onto itself is given by (\theta=\frac{360^{\circ}}{k}), where (k = 1,2,\cdots,n).

Step2: Identify the number of sides of the polygon

The given polygon is an octagon, so (n = 8).

Step3: Calculate the angles of rotation

When (k = 1), (\theta=\frac{360^{\circ}}{1}=360^{\circ}); when (k = 2), (\theta=\frac{360^{\circ}}{2}=180^{\circ}); when (k = 3), (\theta=\frac{360^{\circ}}{3}=120^{\circ}); when (k = 4), (\theta=\frac{360^{\circ}}{4}=90^{\circ}); when (k = 5), (\theta=\frac{360^{\circ}}{5}=72^{\circ}); when (k = 6), (\theta=\frac{360^{\circ}}{6}=60^{\circ}); when (k = 7), (\theta=\frac{360^{\circ}}{7}); when (k = 8), (\theta=\frac{360^{\circ}}{8}=45^{\circ}). The angles of rotation that map the octagon onto itself are (45^{\circ},90^{\circ},135^{\circ},180^{\circ},225^{\circ},270^{\circ},315^{\circ},360^{\circ}) and their multiples.

Answer:

The angles of rotation that carry the octagon onto itself are (45^{\circ},90^{\circ},135^{\circ},180^{\circ},225^{\circ},270^{\circ},315^{\circ},360^{\circ}) and their multiples.