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Question
In the given figure, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that `square`PQRS is a parallelogram.
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Solution
Given: `square` ABCD is a parallelogram.
AP = BQ = CR = DS
To prove: `square`PQRS is a parallelogram.
Proof:
AP = CR ...(Given) ...(i)
`square`ABCD is a parallelogram.
AB = CD ...(opposite sides of parallelogram)
∴ AP + PB = CR + RD ...[A-P-B, D-R-C] ...(ii)
∴ PB = RD ...[From (i) and (ii)] ...(iii)
∠ABC ≅ ∠ADC ...(Opposite angles of parallelogram)
That is, ∠PBQ ≅ ∠RDS ...(A-P-B, B-Q-C, C-R-D and A-S-D) ...(iv)
In ΔPBQ and ΔRDS,
Seg PB ≅ Seg RD ...[From (iii)]
∠PBQ ≅ ∠RDS ...[From (iv)]
Seg BQ ≅ Seg SD ...(Given)
∴ ΔPBQ ≅ ΔRDS ...[SAS test]
Seg PQ ≅ Seg RS ...(c.s.c.t) ...(v)
Thus, we can prove that, ΔPAS ≅ ΔRCQ,
∴ Seg PS ≅ Seg RQ ...(vi)
In `square`PQRS,
Seg PQ ≅ Seg RS ...[From (v)]
Seg PS ≅ Seg RQ ...[from (vi)]
If the opposite sides of a quadrilateral are congruent, then it is a parallelogram.
∴ `square`PQRS is a parallelogram.
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