Question

Answer the following questions to summarize the rotations that carry a regular penagon onto itself. (2 points) A regular pentagon can make turns onto itself before coming around full circle. Each turn would be a multiple of degrees. Question 2 a. What are the degrees of the turns that will allow it to turn on itself?

Answer 1

To determine the degrees of the turns that will allow a regular pentagon to rotate onto itself, we need to consider the symmetry of the pentagon.

A regular pentagon has 5 sides, and the angles of rotation that map the pentagon onto itself are multiples of the angle formed by the vertices when divided evenly around a circle. The full circle is \(360^\circ\).

To find the angles, we can divide \(360^\circ\) by the number of sides of the pentagon (which is 5):

\[
\text{Angle of rotation} = \frac{360^\circ}{5} = 72^\circ
\]

The possible rotations that map the pentagon onto itself are:

1. \(0^\circ\) (no rotation)
2. \(72^\circ\) (one vertex moves to the position of the next vertex)
3. \(144^\circ\) (two vertices move to the next vertex's position)
4. \(216^\circ\) (three vertices move to the next vertex's position)
5. \(288^\circ\) (four vertices move to the next vertex's position)

Thus, the degrees of the turns that will allow the regular pentagon to turn onto itself are:

- \(0^\circ\)
- \(72^\circ\)
- \(144^\circ\)
- \(216^\circ\)
- \(288^\circ\)