In a parallelogram; opposite sides are equal, opposite angles are equal and diagonals bisect each other.
Prove that in a parallelogram:
(i) the opposite sides are equal;
(ii) the opposite angles are equal;
(iii) diagonals bisect each other.
Proof:
Let PQRS be a parallelogram. Draw its diagonal PR.
In ∆ PQR and ∆ RSP,
∠1 = ∠4 (alternate angles)
∠3 = ∠2 (alternate angles)
and PR = RP (common)
Therefore, ∆ PQR ≅ ∆ RSP (by ASA congruence)
⇒ PQ = RS, QR = SP and ∠Q
= ∠S.
Similarly, by drawing the diagonal QS, we can prove that
∆ PQS ≅
∆ RSQ
Therefore, ∠P
= ∠R
Thus, PQ = RS, QR = SP, ∠Q
= ∠S
and ∠P
= ∠R.
This proves (i) and (ii)
In order to prove (iii) consider parallelogram PQRS and draw its diagonals PR
and QS, intersecting each other at O.
In ∆ OPQand ∆ ORS, we have
PQ = RS [Opposite sides of a parallelogram]
∠POQ = ∠ ROS [Vertically opposite angles]
∠OPQ = ∠ORS [Alternate angles]
Therefore, ∆ OPQ ≅ ∆ ORS [By ASA property]
⇒ OP = OR and OQ = OS.
This shows that the diagonals of a parallelogram bisect each other.
The converse of the above result is also true, i.e.,
(i) If the opposite sides of a quadrilateral are equal then it is a parallelogram.
(i) If the opposite angles of a quadrilateral are equal then it is a parallelogram.
(i) If the diagonals of a quadrilateral bisect each other then it is a parallelogram.
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