P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Prove that DA = AR and CQ = QR.

Explanation

Given In a parallelogram ABCD, P is the mid-point of DC.
To prove DA=AR and CQ=QR
Proof ABCD is a parallelogram.

`therefore" "BC=AD and BC ||AD`
Also, `" " DC =AB and DC||AB`
Since, P is the mid-point of DC.
`therefore" "DP =PC=(1)/(2)DC`
Now, `" "QC||AP and PC||AQ`
So, APCQ is a parallelogram.
`therefore" "AQ=PC=(1)/(2)DC`
`" "=(1)/(2)AB=BQ" "[becauseDC=AB]...(i)`
Now, in `Delta`AQR and `Delta`BQC, AQ=BQ`" "` [from Eq. (i)]
`" "angleAQR=angleBQC" "`[vertically opposite angles]
and `" "angleARQ=angleB CQ" "`[Alternate interior angles]
`therefore" "DeltaAQRcongDeltaBQC" "` [by AAS congruence rule] ltBrgt `therefore" " AR=BC" "` [by CPCT rule]
But `" "BC=DA`
`therefore" "AR=DA`
Also, ` " "CQ=QR" "` [ by CPCT rule]
Hence proved.