In 3–5 sentences, explain how a regular pentagon rotations onto itself onto itself. State all possibilities and show work to justify your answer.

Answer 1

A regular pentagon can rotate onto itself in five distinct ways. The rotations occur at angles of \(0^\circ\), \(72^\circ\), \(144^\circ\), \(216^\circ\), and \(288^\circ\).

1. \(0^\circ\) rotation is the identity, where the pentagon remains unchanged.
2. \(72^\circ\) rotation moves each vertex to the position of the next vertex clockwise.
3. \(144^\circ\) rotation shifts each vertex two positions clockwise.
4. \(216^\circ\) rotation moves each vertex three positions clockwise.
5. \(288^\circ\) rotation shifts each vertex four positions clockwise.

These rotations can be verified by noting that each vertex will land on a vertex of the pentagon, thus maintaining the shape and symmetry. This symmetry implies that the regular pentagon has five rotational symmetries.