ABCD is a trapezium in which AB||DC and AD = BC. If P, Q, R, S be respectively the mid points of BA, BD, CD and CA, then PQRS is a

The given figure is a trapezium ABCD with AB parallel to DC and AD equal to BC. P, Q, R, and S are the midpoints of BA, BD, CD, and CA respectively.

Step 1: Since P and Q are midpoints of BA and BD respectively, BQ is half of BD and BP is half of BA.

Step 2: Similarly, since R and S are midpoints of CD and CA respectively, CR is half of CD and CS is half of CA.

Step 3: Since AD = BC, and P and S are midpoints of AD and BC respectively, PS is half of AD and therefore also half of BC.

Step 4: Similarly, since AB = DC, and Q and R are midpoints of AB and DC respectively, QR is half of AB and therefore also half of DC.

Step 5: Since PS and QR are both half of the same length (BC and DC respectively), PS = QR.

Step 6: Since BP = BQ and CR = CS, PQ = RS.

Step 7: Therefore, PQRS is a parallelogram because opposite sides are equal (PS = QR and PQ = RS).