Q. 234.0( 5 Votes )
P is the mi
To Prove: DA = AR and CQ = QR
Given: P is the mid-point of the CD. A line through C parallel to PA intersects AB at Q and DA produced to R.
Opposite sides of a parallelogram are parallel and equal.
ASA Theorem: If two angles and one side of a triangle is equal to two angles and one side of another triangle, then the triangles are congruent.
BC = AD [Opposite sides of a parallelogram]
BC || AD [Opposite sides of a parallelogram]
DC = AB and DC || AB [ Opposite sides of a parallelogram]
Now, it is given that P is mid-point of the CD.
DP = PC = 1/2 DC
QC || AP and PC || AQ,
As the opposite sides are equal and parallel.
APCQ is a parallelogram.
AQ = PC = 1/2 DC = 1/2 AB = BQ
Now, in ΔAQR and BQC,
AQ = BQ
∠AQR = ∠BQC [Vertically opposite angles]
∠ARQ = ∠BCQ [Alternate angles]
Therefore, ΔAQR and BQC are congruent by ASA theorem.
AR = BC [By C.P.C.T]
BC = DA
AR = DA
And, CQ = QR
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