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Ques no. 45
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Dear Student,


1) To prove,PQ∥MN∥RSUsing line through M and parallel to PQ passes through the PS.Corrosponding to △PSQ.Since M is mid point of SQ,  A will also be mid point of SP. (AM∥PQ)SMMQ=SAAP=1 Therefore SA =APsimilarly we can prove B to be the mid point of QR.Now , the segment connecting the mid point of the non-parallel side of the trapizium will be parallel to the parallel side of the trapexium.Therefore,PQ∥MN∥RS.2 )AlsoAB=12PQ-RSMN=AB-MB-NABy the mid segmeorem applied to the △PSR we get, NA=12RSBy the mid segmeorem applied to the △QSR we get, MB=12RSMN=12PQ+RS-12RS-12RS=12PQ-RS

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Sorry ques no. 5 on this
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4) Given: PQRS is a parallelogram , PM=1/3PS , RN=1/3QR
To prove:PNRM is a parallelogram
Construction : join PN and MR
Proof: PS=QR (opposite sides equal)
1/3PS=1/3QR
PM=RN ..(i)

PM||RN ..(ii)

From (i) and (ii) we get a quadrilateral with opposite sides both equal and parallel
Thus PNRM is a parallelogram

Hence proved



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