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 CW =QR.
Given in a parallelogram ABCD, P is the mid-point of DC.
To prove DA=AR and CQ=QR
Proof ABCD is a parallelogram.
∴BC=AD and BC||AD
Also,DC=AB and DC||AB
Since, P is the mid-point of DC.
Now,QC ∥AP and PC ∥ AQ
So, APCQ is a parallelogram,
=12AB=BQ [∵DC=AB] . . . . . .(i)
Now, in ΔAQR and ΔBQC,AQ=BQ[from Eq. (i)]
∠AQR =∠BQC[vertically opposite angles]
and∠ARQ=∠BCQ [Alternate opposite angles]
∴ΔAQR=ΔBQC [by AAS congruence rule]
∴AR = BC [by CPCT rule]
ButBC = DA
∴ AR = DA
Also,CQ = QR [by CPCT rule]