Parallelogram Definition

Understanding quadrilaterals of Class 8

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:

Rectangle

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.

Do follow NCERT Solutions for Class 8 Maths prepared by expert faculty of Physics Wallah. For additional information related to the subject you can check the Maths Formula section.