Question

question 6 of 25 if pqrs is a rhombus, which statements must be true? check all that apply. a. pt≅rt b. pr≅qs c. ∠pqr is supplementary to ∠qps. d. ∠pqr is congruent to ∠qps. e. ps is parallel to qr. f. pr is perpendicular to qs.

Explanation:

Step1: Recall rhombus properties

A rhombus has diagonals that bisect each other. So, in rhombus \(PQRS\) with diagonals \(PR\) and \(QS\) intersecting at \(T\), \(PT = RT\) and \(QT=ST\). So, statement A is true.

Step2: Analyze diagonal - length relationship

The diagonals of a rhombus are not necessarily equal. So, \(PR\) is not always equal to \(QS\), and statement B is false.

Step3: Consider adjacent - angle relationship

Adjacent angles of a rhombus are supplementary. Since \(\angle PQR\) and \(\angle QPS\) are adjacent angles, \(\angle PQR+\angle QPS = 180^{\circ}\), so statement C is true.

Step4: Analyze non - adjacent angle relationship

Opposite angles of a rhombus are congruent, not adjacent angles. So, \(\angle PQR\) is not congruent to \(\angle QPS\) in general, and statement D is false.

Step5: Recall side - parallel relationship

In a rhombus, opposite sides are parallel. Since \(PS\) and \(QR\) are opposite sides, \(PS\parallel QR\), so statement E is true.

Step6: Analyze diagonal - perpendicularity

The diagonals of a rhombus are perpendicular to each other. So, \(PR\perp QS\), and statement F is true.

Answer:

A. \(PT\cong RT\)
C. \(\angle PQR\) is supplementary to \(\angle QPS\)
E. \(\overline{PS}\) is parallel to \(\overline{QR}\)
F. \(\overline{PR}\) is perpendicular to \(\overline{QS}\)