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:

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.

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.

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