Question

the figure shows quadrilateral pqrs. diagonals $overline{pr}$ and $overline{qs}$ 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 properties

One - pair of opposite sides being parallel and congruent proves a parallelogram. Also, if both pairs of opposite sides are congruent or if the diagonals bisect each other, the quadrilateral is a parallelogram.

Step2: Analyze option A

Just $\overline{PS}\cong\overline{QR}$ alone is not enough to prove it's a parallelogram.

Step3: Analyze option B

Just $\overline{PS}\parallel\overline{QR}$ alone is not enough to prove it's a parallelogram.

Step4: Analyze option C

If $\overline{PS}\parallel\overline{QR}$ and $\overline{PS}\cong\overline{QR}$, then by the definition of a parallelogram (a quadrilateral with one - pair of opposite sides parallel and congruent), $PQRS$ is a parallelogram.

Step5: Analyze option D

If $\overline{PQ}\cong\overline{SR}$ and $\overline{PS}\cong\overline{QR}$, then by the parallelogram property (both pairs of opposite sides are congruent), $PQRS$ is a parallelogram.

Step6: Analyze option E

$\overline{PQ}\cong\overline{PS}$ and $\overline{RQ}\cong\overline{RS}$ do not prove that $PQRS$ is a parallelogram. These are adjacent - side congruences, not opposite - side ones.

Step7: Analyze option F

If $\overline{PT}\cong\overline{TR}$ and $\overline{QT}\cong\overline{TS}$, then the diagonals of the quadrilateral bisect each other. By the parallelogram property (diagonals bisect each other), $PQRS$ is a parallelogram.

Answer:

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