Question

A point P is taken on side CD of a parallelogram ABCD and CD is produced to Q making DQ=CP. The

line through Q parallel to AD meets BP produced as S and AD is produced to meet BS at R. Prove

that ARSQ is a parallelogram.​

Answer 1

AQRS  is a parallelogram  point P is taken on side CD of a parallelogram ABCD and CD is produced to Q making DQ=CP

Step-by-step explanation:

in ΔQPS & Δ ABR

QP = AB  ( QP = DQ + DP = CP + DP = DC = AB)

QS ║ AD

QP ║ AB  ( as CD is extended to Q)

=> ΔQPS ≅ Δ ABR

QS = AR

PS = BR

PR + RS = BP + PR

=> BP = RS

QP = AB  

QP ║ AB

=> ABPQ is a parallelogram

=> AQ= BP

=> AQ = RS

now in AQRS

QS = AR

AQ = RS

QS ║ AR

=> AQRS  is a parallelogram

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