Question

in addition to the information given in the drawing, which statement is sufficient to prove pqrs is a parallelogram?

a ( angle qpr cong angle srp )
b ( v ) is the midpoint of ( overline{pr} )
c ( overline{qr} cong overline{sp} )
d ( overline{qp} cong overline{sr} )

Explanation:

Step1: Recall parallelogram properties

A quadrilateral is a parallelogram if one pair of opposite sides is both parallel and congruent.

Step2: Analyze option C

If \(\overline{QR}\cong\overline{SP}\), and from the angles in the drawing (assuming some implicit parallel - like relationships from the angle markings, for example, if \(\angle Q\) and \(\angle S\) are related in a way that can imply \(QR\parallel SP\) (using alternate - interior angles or corresponding angles for parallel lines)), then by the property of a parallelogram (one pair of opposite sides is parallel and congruent), \(PQRS\) is a parallelogram.

Step3: Analyze option A

\(\angle QPR\cong\angle SRP\) only gives information about angles related to the transversal \(PR\) but not enough to prove the whole quadrilateral is a parallelogram.

Step4: Analyze option B

\(V\) being the mid - point of \(\overline{PR}\) only gives information about a segment bisector, not about the sides of the quadrilateral \(PQRS\) being parallel or congruent.

Step5: Analyze option D

\(\overline{QP}\cong\overline{SR}\) alone (without information about their parallelism) is not sufficient to prove \(PQRS\) is a parallelogram.

Answer:

C. \(\overline{QR}\cong\overline{SP}\)