In Fig.9.29, ar (DRC) = ar (DPC) and ar (BDP) = ar (ARC). Show that both the quadrilaterals ABCD and DCPR are trapeziums

We have to prove quadrilateral ABCD and DCPR are trapeziums. 

It is given that Area (ΔDRC) = Area (ΔDPC)

As ΔDRC and ΔDPC are lying on the same base DC having equal areas, therefore, they must lie between the same parallel lines.

According to Theorem 9.3: Two triangle having the same base (or equal bases) and equal areas lie between the same parallels.

∴ DC || RP

Therefore, DCPR is a trapezium 

It is also given that Area (ΔBDP) = Area (ΔARC)

Now, subtract ar (ΔDPC) form ar (ΔBDP) and ar (ΔDRC) from ar (ΔARC) 

ar (ΔBDP) - ar (ΔDPC) = ar (ΔARC) - ar (ΔDRC) [Since, ar (ΔDPC) = ar (ΔDRC)]

ar (ΔBDC) = ar (ΔADC)

Since ΔBDC and ΔADC are on the same base CD having equal areas, they must lie between the same parallel lines. (According to Theorem 9.3)

∴ AB || CD

Therefore, ABCD is a trapezium.