Question

the figure shows quadrilateral pqrs. diagonals pr and qs intersect at point t. choose all the conditions that prove that pqrs is a parallelogram. a. (overline{ps}congoverline{qr}) b. (overline{ps}paralleloverline{qr}) c. (overline{ps}paralleloverline{qr}) and (overline{ps}congoverline{qr}) d. (overline{pq}congoverline{sr}) and (overline{ps}congoverline{qr}) e. (overline{pq}congoverline{ps}) and (overline{rq}congoverline{rs}) f. (overline{pt}congoverline{tr}) and (overline{qt}congoverline{ts})

Explanation:

Step1: Recall parallelogram - definition and properties

A quadrilateral is a parallelogram if: 1. Opposite sides are parallel and equal. 2. Diagonals bisect each other.

Step2: Analyze option A

Just $PS\cong QR$ (option A) is not enough. A quadrilateral with one - pair of equal sides is not necessarily a parallelogram.

Step3: Analyze option B

Just $PS\parallel QR$ (option B) is not enough. A quadrilateral with one - pair of parallel sides is a trapezoid, not necessarily a parallelogram.

Step4: Analyze option C

If $PS\parallel QR$ and $PS\cong QR$, then one pair of opposite sides of the quadrilateral $PQRS$ is parallel and equal. By the definition of a parallelogram, this is a sufficient condition for $PQRS$ to be a parallelogram.

Step5: Analyze option D

If $PQ\cong SR$ and $PS\cong QR$, then both pairs of opposite sides of the quadrilateral $PQRS$ are equal. This is a well - known property of a parallelogram.

Step6: Analyze option E

$PQ\cong PS$ and $RQ\cong RS$ do not satisfy any of the parallelogram properties. These equalities do not imply that $PQRS$ is a parallelogram.

Step7: Analyze option F

If $PT\cong TR$ and $QT\cong TS$, then the diagonals of the quadrilateral $PQRS$ bisect each other. This is a sufficient condition for $PQRS$ to be a parallelogram.

Answer:

C. $PS\parallel QR$ and $PS\cong QR$
D. $PQ\cong SR$ and $PS\cong QR$
F. $PT\cong TR$ and $QT\cong TS$