Question

1. trapezoid abcd was rotated 180 about the origin to create trapezoid pqrs, as shown below. which statement about the trapezoids is true? a. angle a is congruent to angle s. b. angle c is congruent to angle q. c. side bc is congruent to side qr. d. side dc is congruent to side rq.

Explanation:

Step1: Recall rotation property

A 180 - degree rotation about the origin is a rigid transformation. Rigid transformations preserve side - lengths and angle - measures. Corresponding parts of the pre - image (trapezoid \(ABCD\)) and the image (trapezoid \(PQRS\)) are congruent. The order of vertices in a polygon matters for identifying corresponding parts. When a figure is rotated 180 degrees about the origin, the vertices of the pre - image and image are paired up such that \(A\) corresponds to \(S\), \(B\) corresponds to \(R\), \(C\) corresponds to \(Q\), and \(D\) corresponds to \(P\).

Step2: Analyze each option

• Option A: Angle \(A\) corresponds to angle \(S\), but they are not congruent in the sense of the trapezoid's internal angle relationships in the way the question implies.
• Option B: Angle \(C\) corresponds to angle \(Q\), but they are not congruent in the non - corresponding way stated.
• Option C: Side \(BC\) does not correspond to side \(QR\).
• Option D: Since trapezoid \(ABCD\) is rotated 180 degrees about the origin to get trapezoid \(PQRS\), side \(DC\) corresponds to side \(RQ\). Corresponding sides of a figure and its 180 - degree rotation about the origin are congruent.

Answer:

D. Side \(DC\) is congruent to side \(RQ\)