Given: ABCD is a parallelogram. P and Q are the midpoints of sides AB and DC respectively.
To prove: APCQ is a parallelogram.
Proof:
AP = (1/2) AB …..(i) [P is the midpoint of side AB]
QC = (1/2) DC ….(ii) [Q is the midpoint of side CD]
ABCD is a parallelogram. [Given]
∴ AB = DC [Opposite sides of a parallelogram]
∴ (1/2) AB = (1/2) DC [Multiplying both sides by 1/2]
∴ AP = QC ….(iii) [From (i) and (ii)]
Also, AB || DC [Opposite angles of a parallelogram]
i.e. AP || QC ….(iv) [A – P – B, D – Q – C]
From (iii) and (iv),
APCQ is a parallelogram. [A quadrilateral is a parallelogram if its opposite sides is parallel and congruent]
