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P and Q are the mid-point of the opposit...

P and Q are the mid-point of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PQRS is a parallelogram.

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Given: A parallelogram ABCD in which P and Q are the mid-points of the sides AB and CD respectively. AQ intersects DP at S and BQ interects CP respectively. AQ intersects DP at S and BQ intersects Cp at R.
To prove: PRQS is a parallelogram.
`"Proof:"" "DC"||"AB" (because""opposite sides of a parallelogram are parallel")`
`implies" "AP"||"QC`
`" "DC"||"AB" "(because"opposite sides of a parallelogram are equal")`
`implies" "1/2DC=1/2AB`
`implies" "QC=AP" "(because"P is mid-point of AB and Q is mid-points of CD")`
`implies" "APCQ"is a parallelogram."" "(becauseAP"||""QCand QC=AP)`
`therefore" "AQ"||"PC" "(because"opposite sides of a parallelogram are parallel")`
`implies" "SQ"||"PR"`
`"Similarly,"" "SP"||"QR`
`therefore` Quadrilateral PQRS is a parallelogram.
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