In the given figure, if PQRS is a parallelogram and AB || PS, then prove that OC || SR.


Given,

PQRS is a parallelogram,


So, PQ || SR and PS || QR.


Also, AB || PS.



To prove OC || SR


In ∆OPS and OAB,


PS | | AB


POS = AOB [common angle]


OSP = OBA [corresponding angles]


∆OPS ∆OAB [by AAA similarity criteria]


Then,


PS/AB = OS/OB …(i) [by basic proportionality theorem]


In ∆CQR and ∆CAB,


QR || PS || AB


QCR = ACB [common angle]


CRQ = CBA [corresponding angles]


∆CQR ∆CAB


Then, by basic proportionality theorem



[PS QR Since, PQRS is a parallelogram,]


From Equation (i) and (ii),



On subtracting from both sides, we get,



By converse of basic proportionality theorem, SR || OC


Hence proved.


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