Answer the following questions to summarize the rotations that carry a regular penagon onto itself. (4 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? b. Since a regular pentagon can make turns less than 360 degrees onto itself, it has what kind of symmetry? (2 points)

Answer 1

To summarize the rotations that carry a regular pentagon onto itself:

Question 2:

a. The degrees of the turns that will allow a regular pentagon to turn onto itself are multiples of \( \frac{360}{5} = 72 \) degrees. Therefore, the possible rotations are:

• \( 0 \) degrees (no rotation),
• \( 72 \) degrees (1 vertex to the position of the next vertex),
• \( 144 \) degrees (2 vertices),
• \( 216 \) degrees (3 vertices),
• \( 288 \) degrees (4 vertices).

In total, the valid rotations that map the pentagon onto itself are \( 0 \), \( 72 \), \( 144 \), \( 216 \), and \( 288 \) degrees.

b. Since a regular pentagon can make turns less than \( 360 \) degrees onto itself, it has rotational symmetry. This type of symmetry allows an object to appear unchanged after certain rotations.