Question

the figure shows quadrilateral pqrs. diagonals $overline{pr}$ and $overline{qs}$ intersect at point $t$. choose all the statements that contain enough information to prove that pqrs is a parallelogram. a. $\triangle pqrcong\triangle rsp$ b. $overline{ps}congoverline{qr}$ and $overline{pq}paralleloverline{sr}$ c. $angle spqcongangle qrs$ and $angle rspcongangle pqr$ d. $angle stpcongangle qtr$ and $angle ptqcongangle rts$

Explanation:

Step1: Recall parallelogram - related theorems

If two - triangles formed by the diagonals or sides of a quadrilateral are congruent, or if opposite sides are congruent and parallel, or if opposite angles are congruent, the quadrilateral is a parallelogram.

Step2: Analyze option A

If $\triangle PQR\cong\triangle RSP$, then by the congruence of triangles, we can get corresponding sides and angles equal. For example, $PQ = SR$ and $PR=SP$, and the included - angles are equal. This implies that opposite sides are equal and parallel (by using properties of congruent triangles), so the quadrilateral $PQRS$ is a parallelogram.

Step3: Analyze option B

If $\overline{PS}\cong\overline{QR}$ and $\overline{PQ}\parallel\overline{SR}$, one pair of opposite sides is congruent and parallel. According to the parallelogram theorem, a quadrilateral with one pair of opposite sides congruent and parallel is a parallelogram.

Step4: Analyze option C

If $\angle SPQ\cong\angle QRS$ and $\angle RSP\cong\angle PQR$, the opposite angles of the quadrilateral are congruent. By the property that a quadrilateral with opposite angles congruent is a parallelogram, $PQRS$ is a parallelogram.

Step5: Analyze option D

$\angle STP\cong\angle QTR$ and $\angle PTQ\cong\angle RTS$ only tell us about the vertical - angles formed by the diagonals. This information is not sufficient to prove that the quadrilateral $PQRS$ is a parallelogram.

Answer:

A. $\triangle PQR\cong\triangle RSP$
B. $\overline{PS}\cong\overline{QR}$ and $\overline{PQ}\parallel\overline{SR}$
C. $\angle SPQ\cong\angle QRS$ and $\angle RSP\cong\angle PQR$