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.

**Parallelogram**

**Properties of a Rectangle Rhombus and Square**

**Practice Test on Parallelogram**

**Parallelogram - Worksheet**

**8th Grade Math Practice**

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