Question

examine the following paragraph proof.
consider parallelogram pqrs, where \\(\overline{pq} \cong \overline{rs}\\) and \\(\overline{qr} \cong \overline{ps}\\). construct diagonal pr and diagonal qs. by the sss criterion, \\(\triangle pqr \cong \triangle rsp\\) and \\(\triangle pqs \cong \triangle rsq\\). therefore, \\(\angle qps \cong \angle srq\\) and \\(\angle pqr \cong \angle rsp\\) since corresponding angles of congruent triangles are congruent.
what statement is proven by the given steps?

a. opposite angles of a parallelogram are congruent.

b. consecutive angles of a parallelogram are congruent.

c. a quadrilateral with congruent consecutive angles is a parallelogram.

d. a quadrilateral with congruent opposite angles is a parallelogram.

Explanation:

1. The proof starts with a parallelogram \(PQRS\) and constructs diagonals to form congruent triangles (\(\triangle PQR \cong \triangle RSP\) and \(\triangle PQS \cong \triangle RSQ\)) using the SSS criterion.
2. From the congruent triangles, it concludes that corresponding angles (\(\angle QPS \cong \angle SRQ\) and \(\angle PQR \cong \angle RSP\)) are congruent. These are opposite angles of the parallelogram.
3. Option B is incorrect because consecutive angles in a parallelogram are supplementary, not congruent. Options C and D are incorrect because the proof starts with a parallelogram and proves a property of its angles, not a condition for a quadrilateral to be a parallelogram.

Answer:

A. Opposite angles of a parallelogram are congruent.