The given figure shows parallelograms ABCD and APQR. Show that these parallelograms are equal in area.

We know that,

Opposite sides of || gm are equal and parallel.

∴ AB || DC and AR || PQ.

We know that,

The area of triangle is half that of a parallelogram on the same base and between the same parallels.

From figure,

|| gm ABCD and △ ABR lies on same base AB and between same parallel lines AB and DC.

∴ Area of △ ABR = $\dfrac{1}{2}$ Area of || gm ABCD

⇒ Area of || gm ABCD = 2 Area of △ ABR …….(1)

We know that,

Area of triangles on the same base and between the same parallels lines are equal.

△ ABR and △ APR lie on same base AR and between same parallel lines AR and PQ.

∴ Area of △ ABR = Area of △ APR ……..(2)

From equations (1) and (2), we get :

⇒ Area of || gm ABCD = 2 Area of △ APR ………(3)

Also, || gm APQR and △ APR lies on same base AR and between same parallel lines AR and PQ.

∴ Area of △ APR = $\dfrac{1}{2}$ Area of || gm APQR …….(4)

Using value of area of △ APR from equation (4) in (3), we get :

⇒ Area of || gm ABCD = $2 \times \dfrac{1}{2}$ Area of || gm APQR

⇒ Area of || gm ABCD = Area of || gm APQR.

Hence, proved that the parallelograms ABCD and APQR are equal in area.