Consider △ MCQ and △ MPB
From the figure we know that
∠ QCM and ∠ PBM are alternate angles
So we get
∠ QCM = ∠ PBM
We know that M is the midpoint of BC
CM = BM
∠ CMQ and ∠ PBM are vertically opposite angles
∠ CMQ = ∠ PBM
By ASA congruence criterion
△ MCQ ≅ △ MPB
So we get
Area of △ MCQ = Area of △ MPB
We know that
Area of ABCD = Area of APQD + Area of DMPB – Area of △ MCQ
So we get
Area of ABCD = Area of APQD
Therefore, it is proved that ar (ABCD) = ar (APQD).