Question

if pqrs is a rhombus, which statements must be true? check all that apply. a. $overline{ps}$ is parallel to $overline{qr}$ b. $overline{pr}$ is perpendicular to $overline{qs}$ c. $angle pqr$ is supplementary to $angle qps$ d. $pr = qs$ e. $angle pqr$ is congruent to $angle qps$

Explanation:

Step1: Recall rhombus properties

A rhombus is a parallelogram with all - sides equal. In a parallelogram, opposite sides are parallel. So, in rhombus \(PQRS\), \(\overline{PS}\parallel\overline{QR}\) and \(\overline{PQ}\parallel\overline{RS}\).

Step2: Recall diagonal properties

The diagonals of a rhombus are perpendicular to each other. So, \(\overline{PR}\perp\overline{QS}\).

Step3: Recall adjacent - angle properties

In a parallelogram (and thus in a rhombus), adjacent angles are supplementary. So, \(\angle PQR+\angle QPS = 180^{\circ}\), which means \(\angle PQR\) is supplementary to \(\angle QPS\).

Step4: Analyze diagonal lengths

The diagonals of a rhombus are not necessarily equal. Only in a square (a special - case of a rhombus) are the diagonals equal.

Step5: Analyze angle - congruence

In a rhombus, opposite angles are congruent, not adjacent angles. So, \(\angle PQR\) is not congruent to \(\angle QPS\) (in general, unless it's a square).

Answer:

A. \(\overline{PS}\) is parallel to \(\overline{QR}\)
B. \(\overline{PR}\) is perpendicular to \(\overline{QS}\)
C. \(\angle PQR\) is supplementary to \(\angle QPS\)