Q3 P and Q are any two points lying on the sides DC and AD respectively of a parallelogram ABCD. Show that
Solution:
Given,
P and Q are any two points lying on the sides DC and AD respectively of a parallelogram ABCD.
and parallelogram ABCD are on the same base AB and between the same parallel lines AB and DC. Therefore, (Parallelogram ABCD)……………equation (1)

Similarly, and parallelogram ABCD are on the same base BC and these are between the same parallel lines AD and BC

Therefore, (Parallelogram ABCD)……………..equation (2)

From (1) and (2), we have
Q4 In the figure is a point in the interior of a parallelogram ABCD. Show that


Solution (i)
A line GH is drawn parallel to AB passing through P.In a parallelogram, (By construction)……….equation (1)
Thus,

and

……….equation (2)
From equations (1) and (2),


is a parallelogram.
Now,
In and parallelogram ABHG are lying on the same base AB and between the same parallel lines AB and GH. Therefore, ………….equation (3)
Also,
In and parallelogram CDGH are lying on the same base CD and between the same parallel lines CD and GH.
…………equation (4)
Adding equations (3) and (4),
Solution (ii)
A line EF is drawn parallel to AD passing through P.
In a parallelogram, (by construction)…………equation (6)
Thus, ……….equation (7)
From equations (6) and (7),


AEFD is a parallelogram.
Now,
In ΔAPD and parallelogram AEFD are lying on the same base AD and between the same parallel lines AD and EF, ………equation (8)
Also, in and parallelogram BCFE are lying on the same base BC and between the same parallel lines BC and EF ……..equation (9)
Adding equations (8) and (9),
From equation (5) and (10)