Given In a parallelogram ABCD, P and Q are the mid-points of AB and CD, respectively.

To show PRQS is a parallelogram.

Proof Since, ABCD is a parallelogram.

`" "AB||CD" "`

`rArr" "AP||QC`

Also, `" "AB=DC`

`" "(1)/(2)AB=(1)/(2)DC" "` [dividing both sides by 2]

`rArr" "AP=QC" "` [ since, P and Q are the mid-points of AB and DC]

Now, `" "AP||QC and AP=QC`

Thus APCQ is a parallelogram.

`therefore" "AQ||PC or SQ||PR" "...(i)`

Again, `" "AB||DC or BP||DQ`

Also, `" "AB=DC rArr(1)/(2)AB=(1)/(2)DC" "` [dividing both sides by 2]

`rArr" "BP=QD` [since, P and Q are the mid-points of AB anc DC]

Now, `" "BP||QD and BP=QD`

So, BPDQ is a parallelogram.

`therefore" "PD||BQ or PS||QS" "...(iii)`

From Eqs. (i) and (ii), `" "SQ||RQ and PS||QR`

So, PRQS is a parallelogram. `" "` Hence proved.