In the given figure, if PQRS is a parallelogram and AB || PS, then pro

1.	In the given figure, if PQRS is a parallelogram and AB \|\| PS, then prove that OC \|\| SR.
Answer» To prove - OC║SRProof - In ΔOPS and ΔOAB∠POS = ∠AOB (common in both)∠OSP = ∠OBA (corresponding angles are equal as PS║AB)=> ΔOPS ~ ΔOAB [AA criteria]=> PS/AB = OS/OB ........................(1) (sides in similar triangles are proportional)In ΔCAB and ΔCRQAs, QR║AB=> ∠QCR = ∠ACB (common)=> ∠CBA = ∠CRQ (corresponding angles are equal)=> ΔCAB ~ ΔCQR [AA criteria]=> CR/CB = QR/AB (sides in similar triangles are proportional)Also, PS = QR [ PQRS is parallelogram]=> CR/CB = PS/AB ......................(2)From (1) and (2)=> OS/OB = CR/CB=> OB/OS = CB/CRSubtracting 1 from both sidesSo, OB/OS - 1 = CB/CR - 1=> (OB - OS)/OS = (CB - CR)/CR=> BS/OS = BR/CRBy converse of Thales Theorem=> OC║SR . Hence proved .

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