Question

1. a quadrilateral has rotation symmetry that can take any of its vertices to any of its other vertices.

select all conclusions that we can reach from this.
a. all sides of the quadrilateral have the same length.
b. all angles of the quadrilateral have the same measure.
c. all rotations take one half of the quadrilateral to the other half of the quadrilateral.

Explanation:

Step1: Analyze rotation symmetry for sides

Since rotation symmetry can take any vertex to any other vertex, the distances between consecutive vertices (sides) must be equal. So all sides have the same length.

Step2: Analyze rotation symmetry for angles

Similarly, the angles between consecutive sides (angles of the quadrilateral) must be equal as the rotation maps one angle - side - angle configuration to another. So all angles have the same measure.

Step3: Analyze option C

Rotation symmetry that maps vertices to vertices does not imply that all rotations take one half of the quadrilateral to the other half. For example, a non - degenerate rotation (not \(180^{\circ}\)) of a square (a quadrilateral with the given rotation symmetry) does not split the square into two halves and map one half to the other.

Answer:

A. All sides of the quadrilateral have the same length.
B. All angles of the quadrilateral have the same measure.