prove that in a parallelogram
(i)the opposite sides are equal
(ii)the opposite angles are equal
(iii)diagonals bisect each other
Answers
Answer:
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.
Step-by-step explanation:
bisect each other -
THEOREM: If a quadrilateral has diagonals which bisect each other, then it is aparallelogram. THEOREM: If a quadrilateral has one set of opposite sides which are both congruent and parallel, then it is a parallelogram. This last method can save time and energy when working a proof!
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