# Parallelogram Definition

## Parallelogram

### Properties of parallelogram are as follows:

 Opposite sides of a parallelogram are equal. If ABCD is a parallelogram then AD = BC, AB = DC. Opposite angles of a parallelogram are equal. If ABCD is a parallelogram ∠A = ∠Cand ∠B = ∠D. The diagonals of a parallelogram bisect each other. If ABCD is a parallelogram then AO = OC and BO = OD.

Area of Parallelogram (A) : Base (b) . Height (h)

Theorem: In a parallelogram, prove that

• the opposite sides are equal;
• the opposite angles are equal;
• diagonals bisect each other.

Proof: Let ABCD be a parallelogram. Draw its diagonal AC.

 In triangles ABC and CDA, we have ∠1 = ∠3 [Alternate angles] ∠2 = ∠4 [Alternate angles] and,    AC = AC [Common side] So, by ASA congruence criterion, we have ΔABC≅ΔCDA ⇒ AB = CD, BC = AD and ∠B = ∠D

Corresponding parts of congruent triangle are equal]

Similarly, by drawing the diagonal BD, we can prove that

ΔABD ≅ΔCDB

⇒ AD = BCand∠A = ∠C

Hence, in parallelogram ABCD, we have

 AB = CD, AD = BC, ∠A = ∠C, ∠B = ∠D This proves (i) and (ii) In order to prove (iii), consider a parallelogram ABCD. Draw its diagonals AC and BD, intersecting each other at O. In Δs AOB and COD, we have

AB = CD [Opposite sides of a parallelogram are equal]

∠AOB = ∠COD [Vertically opposite angles]

∠OAB = ∠DCO [Alternate angles]

∴ ΔOAB≅ΔOCD

⇒ OA = OC and OB = OD

This proves (iii).

### Types of Parallelogram:

#### Recta​ngle

 Rectangle is parallelogram with a right angle. It has the following properties. Each of the angles is a right angle.  Diagonals are equal.

#### Rhombus

Rhombus is a parallelogram whose adjacent sides and adjacent angles are equal. So, rhombus has all the properties of a parallelogram.

 Hence, from the properties of a parallelogram, we have  PQ = SR, PS = RQ  ∠P = ∠R and ∠S = ∠Q  Diagonals PR and SQ bisect each other at right angle  Diagonals are unequal in length.

#### Square

 A square is an equilateral rectangle. This means a square has all the properties of a rectangle with an additional requirement that all the sides have equal length. In square, the diagonals
• bisect one another
• are of equal length
• are perpendicular to one another.

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