Question

the figure shows quadrilateral pqrs. diagonals \\( \overline { p r } \\) and \\( \overline { q s } \\) intersect at point t.
choose all the statements that contain enough information to prove that pqrs is a parallelogram.
a \\( \triangle p q r \cong \triangle r s p \\)
b. \\( \overline { p s } \cong \overline { q r } \\) and \\( \overline { p q } \parallel \overline { s r } \\)
c \\( \angle s p q \cong \angle q r s \\) and \\( \angle r s p \cong \angle p q r \\)
d. \\( \angle s t p \cong \angle q t r \\) and \\( \angle p t q \cong \angle r t s \\)

Explanation:

Step1: Analyze Option A

If \(\triangle PQR\cong\triangle RSP\), then \(PQ = RS\) and \(PR=PR\) (common side). By SSS (Side - Side - Side) congruence, but we can also get \(\angle QPR=\angle SRP\) and \(\angle QRP=\angle SPR\). So \(PQ\parallel RS\) (alternate interior angles) and \(PS\parallel QR\) (alternate interior angles). A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

Step2: Analyze Option B

If \(\overline{PS}\cong\overline{QR}\) and \(\overline{PQ}\parallel\overline{SR}\), a quadrilateral with one pair of opposite sides parallel and equal is a parallelogram.

Step3: Analyze Option C

If \(\angle SPQ\cong\angle QRS\) and \(\angle RSP\cong\angle PQR\), the sum of the interior angles of a quadrilateral is \(360^{\circ}\). So \(\angle SPQ+\angle PQR = 180^{\circ}\) and \(\angle PQR+\angle QRS=180^{\circ}\). Then \(PS\parallel QR\) and \(PQ\parallel RS\) (consecutive interior angles supplementary). A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

Step4: Analyze Option D

If \(\angle STP\cong\angle QTR\) and \(\angle PTQ\cong\angle RTS\), these are just vertical - angle - like statements (but not related to the properties of a parallelogram such as side - side, side - angle - side for proving parallelograms. It does not give information about the sides or other angles to prove that \(PQRS\) 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 \(\angle RSP\cong\angle PQR\)