Final answer:
Measure of angle QPS (m∠QPS) must be equal to the angle opposite to it, or 180 degrees minus its adjacent angle, to ensure PQRS is a parallelogram, utilizing properties such as congruent opposite angles and consecutive angles summing to 180 degrees.
Step-by-step explanation:
To determine what must m∠QPS be for PQRS to be a parallelogram, we must recall the properties of a parallelogram. Specifically, one of the key properties is that opposite angles are congruent (equal). Therefore, m∠QPS must be equal to the angle opposite to it, which is angle R if PQRS is indeed a parallelogram.
Without loss of generality, if we know angle QPS is adjacent to angle QSR, and we know the measure of angle QSR, we could also relate the two angles by considering that consecutive angles in a parallelogram sum up to 180 degrees. This means that m∠QPS = 180° - m∠QSR if angles QPS and QSR are consecutive angles in the parallelogram.
The geometric properties of a parallelogram, such as those involving angles and sides, help ensure the definition of a parallelogram is maintained regardless of transformations like rotations or translations of the coordinate system. These properties are critical in geometric applications and proofs.