Question

what is one of the angles of rotation needed to rotate this octagon onto itself?

Explanation:

Step1: Recall the formula for rotation angles of a regular polygon

The formula for the angle of rotation of a regular polygon is \(\frac{360^{\circ}}{n}\), where \(n\) is the number of sides. For an octagon, \(n = 8\).

Step2: Calculate the minimum rotation angle

\(\frac{360^{\circ}}{8}=45^{\circ}\). Multiples of this angle (\(45^{\circ}\times k\), where \(k = 1,2,\cdots\)) will also map the octagon onto itself.
When \(k = 3\), \(45^{\circ}\times3 = 135^{\circ}\); when \(k = 8\), \(45^{\circ}\times8=360^{\circ}\); when \(k = 2\), \(45^{\circ}\times2 = 90^{\circ}\); when \(k = 6\), \(45^{\circ}\times6=270^{\circ}\); when \(k = 4\), \(45^{\circ}\times4 = 180^{\circ}\); when \(k=5\), \(45^{\circ}\times5 = 225^{\circ}\); when \(k = 7\), \(45^{\circ}\times7=315^{\circ}\); when \(k = 8\), \(45^{\circ}\times8 = 360^{\circ}\). Among the given options \(120^{\circ}\), \(240^{\circ}\), \(135^{\circ}\), the angle that is a multiple of \(45^{\circ}\) is \(240^{\circ}\) (since \(240\div45=\frac{16}{3}\) is wrong, \(120\div45=\frac{8}{3}\) is wrong, \(135\div45 = 3\)).

Answer:

\(135^{\circ}\)