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. (from unit 1, lesson 16.) all sides of the quadrilateral have the same length all angles of the quadrilateral have the same measure. all rotations take one half of the quadrilateral to the other half of the quadrilateral.

Explanation:

• For "All sides of the quadrilateral have the same length":
• Rotation symmetry that maps any vertex to another implies that the distances between vertices (side - lengths) must be equal. Because when we rotate a vertex to another vertex, the side - length (the distance between the original vertex and its adjacent vertex) must be the same as the distance between the new vertex (the image of the original vertex under rotation) and its adjacent vertex.
• For "All angles of the quadrilateral have the same measure":
• Since rotation symmetry can map one vertex to another, the angles at each vertex must be congruent. When we rotate the quadrilateral so that one vertex moves to another, the angle at the original vertex (which is formed by two adjacent sides) must match the angle at the new vertex (formed by the two adjacent sides at the image of the rotated vertex).
• For "All rotations take one half of the quadrilateral to the other half of the quadrilateral":
• This is not necessarily true. Rotation symmetry that maps vertices to vertices does not imply that the quadrilateral is divided into two equal halves in the sense that any rotation would map one "half" to the other. For example, consider a square (which has the given rotation symmetry). A \(90^{\circ}\) rotation does not map one "half" (in a non - symmetrically divided sense) to another. The key property is about vertex - to - vertex mapping and side - length and angle congruence, not arbitrary half - to - half mapping.

Answer:

All sides of the quadrilateral have the same length, All angles of the quadrilateral have the same measure.