Question

a regular pentagon is shown. describe the rotations and the reflections that carry the regular pentagon onto itself. move the correct answer to each box. not all answers will be used. 36° 72° 108° an adjacent vertex an opposite vertex the midpoint of the opposite side a rotation about the center carries the regular pentagon onto itself if and only if the rotation is a multiple of box. a reflection carries the regular pentagon onto itself if and only if the line of reflection contains a vertex and box.

Explanation:

Step1: Determine Rotation Angle

For a regular \( n \)-gon, the smallest rotation angle that maps it onto itself is \( \frac{360^\circ}{n} \). For a pentagon, \( n = 5 \), so \( \frac{360^\circ}{5}=72^\circ \). Rotations that are multiples of \( 72^\circ \) (like \( 72^\circ, 144^\circ, 216^\circ, 288^\circ \)) will carry it onto itself.

Step2: Determine Reflection Line

In a regular pentagon, a reflection line must pass through a vertex and the midpoint of the opposite side (since there's no "opposite vertex" in a pentagon—each vertex has two adjacent vertices and one side opposite, with a midpoint). So the line of reflection contains a vertex and the midpoint of the opposite side.

Answer:

• A rotation about the center carries the regular pentagon onto itself if and only if the rotation is a multiple of \( \boldsymbol{72^\circ} \).
• A reflection carries the regular pentagon onto itself if and only if the line of reflection contains a vertex and \( \boldsymbol{\text{the midpoint of the opposite side}} \).