Final answer:
MQNS is proven to be a parallelogram in parallelogram PQRS by utilizing angle bisectors and the properties of parallelograms that opposite sides are parallel and equal which creates two pairs of congruent triangles.
Step-by-step explanation:
To prove that MQNS is a parallelogram, let's analyze the given conditions in parallelogram PQRS. We know that the bisectors of angles PQR and PSR meet the diagonal at points M and N, respectively. Since PQRS is a parallelogram, opposite sides are parallel and equal, and opposite angles are equal. Angle bisectors divide angles into two equal parts.
MQ and NS are drawn from the angle bisectors to points M and N on diagonal QS of parallelogram PQRS. As MQ and NS are from the angle bisectors of a parallelogram, it means that MQ is parallel to NS, because the angles at P and S are bisected, implying that angles at M and N on QS are corresponding angles, making MQ || NS. Also, since MQ and NS are angle bisectors, the triangles PMQ and SNR formed by these bisectors are congruent by the ASA (Angle-Side-Angle) congruence rule (as angle P = angle S, QS is common, and the angles bisected by MQ and NS are equal). Since these triangles are congruent, sides MQ and NS are equal by the corresponding parts of congruent triangles are equal (CPCTC) criterion.
In the same fashion, MS and NQ can be proved to be parallel and equal by considering the congruence of triangles PMR and SNQ by the ASA rule. Hence, having two pairs of opposite sides that are equal and parallel, we can conclude that MQNS is indeed a parallelogram.