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.

\\(\square\\) a. \\(\triangle pqr \cong \triangle rsp\\)
\\(\square\\) b. \\(\overline{ps} \cong \overline{qr}\\) and \\(\overline{pq} \parallel \overline{sr}\\)
\\(\square\\) c. \\(\angle spq \cong \angle qrs\\) and \\(\angle rsp \cong \angle pqr\\)
\\(\square\\) d. \\(\angle stp \cong \angle qtr\\) and \\(\angle ptq \cong \angle rts\\)

Explanation:

• Option A: If $\triangle PQR \cong \triangle RSP$, then corresponding sides are equal: $PQ=RS$, $QR=SP$, $PR=RP$. A quadrilateral with both pairs of opposite sides congruent is a parallelogram.
• Option B: $\overline{PS} \cong \overline{QR}$ (one pair of sides congruent) and $\overline{PQ} \parallel \overline{SR}$ (one pair of sides parallel) is not sufficient. This could describe an isosceles trapezoid, not necessarily a parallelogram.
• Option C: $\angle SPQ \cong \angle QRS$ and $\angle RSP \cong \angle PQR$ means both pairs of opposite angles are congruent. A quadrilateral with both pairs of opposite angles congruent is a parallelogram.
• Option D: $\angle STP \cong \angle QTR$ and $\angle PTQ \cong \angle RTS$ are just vertical angles, which are always congruent regardless of the quadrilateral type. This provides no unique information to prove it is a parallelogram.

Answer:

A. $\triangle PQR \cong \triangle RSP$
C. $\angle SPQ \cong \angle QRS$ and $\angle RSP \cong \angle PQR$