Final answer:
To show that PRQS is a parallelogram in the given scenario, we use the properties of parallelograms, the Midsegment Theorem, and the Angle-Side-Angle postulate to prove that opposite sides PR and QS as well as PS and QR are parallel, confirming PRQS is a parallelogram.
Step-by-step explanation:
The question asks how to show that PRQS is a parallelogram given that P and Q are midpoints of the opposite sides AB and CD of parallelogram ABCD, with AQ intersecting DP at S and BQ intersecting CP at R.
We know that in a parallelogram, opposite sides are equal and parallel. Since P and Q are midpoints, AP = PB and CQ = QD. By the Midsegment Theorem, which states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length, we can understand that PQ is parallel and equal to both AB and CD, because ABCD is a parallelogram itself. This shows that PQ is a line segment that does not change its length or direction during a rotation.
Now focusing on the intersection points S and R, we observe triangles APD and BQC. Since diagonals of a parallelogram bisect each other, AS = SD and BR = RC. Also by Angle-Side-Angle (ASA) postulate, triangles APD and BQC are congruent which implies that angles APS and BQR are equal, and therefore PS is parallel to QR. Similarly, since triangles APB and CQD are on the same lines, PR is parallel to QS. The fact that opposite sides PR and QS as well as PS and QR are parallel means that PRQS is indeed a parallelogram.