Question

question 19 of 25 if pqrs is a rhombus, which statements must be true? check all that apply. a. ∠pqr is supplementary to ∠qps. b. (overline{ps}) is parallel to (overline{qr}). c. (overline{pt}congoverline{rt}) d. ∠pqr is congruent to ∠qps. e. (overline{pr}) is perpendicular to (overline{qs}). f. (overline{pr}congoverline{qs})

Explanation:

Step1: Recall rhombus properties

A rhombus has opposite - sides parallel. So, $\overline{PS}\parallel\overline{QR}$, making option B true.

Step2: Recall diagonal properties

The diagonals of a rhombus bisect each other. So, if the diagonals $\overline{PR}$ and $\overline{QS}$ intersect at $T$, then $\overline{PT}\cong\overline{RT}$, making option C true.

Step3: Recall diagonal - angle relationship

The diagonals of a rhombus are perpendicular to each other. So, $\overline{PR}\perp\overline{QS}$, making option E true.

Step4: Analyze angle - related options

Adjacent angles of a rhombus are supplementary, but there is no information to suggest that $\angle PQR$ is supplementary to $\angle QPS$ in the way option A states. Also, $\angle PQR$ and $\angle QPS$ are adjacent angles and not congruent (in general for a non - square rhombus), so option D is false. And the diagonals of a rhombus are not equal in length (in general for a non - square rhombus), so $\overline{PR}
eq\overline{QS}$, making option F false.

Answer:

B. $\overline{PS}$ is parallel to $\overline{QR}$, C. $\overline{PT}\cong\overline{RT}$, E. $\overline{PR}$ is perpendicular to $\overline{QS}$