Maths: Quadrailateral?

(a)
ABCD is parallelogram --> BC is parallel to AD, and BC=AD
PADQ is parallelogram --> PQ is parallel to AD, and PQ=AD
BC=AD=PQ

(b)
BC=PQ
BC is parallel to AD, and PQ is parallel to AD ---> PQ is parallel to BC
so PBCQ is parallelogram

(a)
ABCD is a //gram => BC=AD
PADQ is a //gram => PQ=AD
∴ BC = PQ
(b)
ABCD is a //gram => AB // DC
PADQ is a //gram => AP // DQ
∴ taken AD as a transveral through the 2 // lines AP, DQ ; we have :
∠ BAP = ∠ CDQ　　　( 2 // lines, corr.∠s equal ) . . . . . . . . . . . . . (#)

ABCD is a //gram => AB = DC . . . . . . . . . . . . . ①
PADQ is a //gram => AP = DQ . . . . . . . . . . . . . ②

From (#),①,② : ∆BAP ≡ ∆CDQ 　　( SAS )　
　　 　　 　 =>　　BP = CQ
& since : 　　　　　BC = PQ 　　( From result of (a) )
we have :　 PBCQ is a //gram　　( oppo. sides equal )