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Properties of Parallelograms

  • Last Updated : 02 Sep, 2021

A quadrilateral having both the pairs of opposite sides equal is a parallelogram. A parallelogram is a two-dimensional geometrical shape, whose sides are parallel to each other. Below are some simple facts about parallelogram:

  1. Number of sides in Parallelogram = 4
  2. Number of vertices in Parallelogram = 4
  3. Area = Base x Height
  4. Perimeter = 2 (Sum of adjacent sides length)
  5. Type of polygon = Quadrilateral

Below is the representation of a parallelogram:

Proofs: Parallelograms

Proof 1: Opposite sides of a parallelogram is equal.

Given: ABCD is a parallelogram
 



To Prove: AB = CD & DA = BC

Firstly, Join AC

As given ABCD is a parallelogram. Therefore, 

AB || DC  &  AD || BC

Now,  AD || BC and AC is intersecting A and C respectively.

\angle      DAC = \angle      BCA                   …(i)                  [Alternate Interior Angles]

Now, AB || DC and AC is intersecting A and C respectively.

\angle      BAC = \angle      DCA                 …(ii)                      [Alternate Interior Angles]



Now, In \triangle      ADC & \triangle    CBA

\angle      DAC = \angle      BCA                    [ From (i) ]

AC = AC                                   [ Common Side ]

\angle      DCA = \angle      BAC                 [ From (ii) ]

So, by ASA(Angle-Side-Angle) criterion of congruence

\triangle      ADC  \cong      \triangle     CBA

AB = CD & DA = BC [ Corresponding part of congruent triangles are equal ]

Hence Proved !

Proof 2: Opposite angles of a parallelogram are equal.

Given: ABCD is a parallelogram



To Prove:  \angle      A = \angle      C  and \angle      B = \angle      D

As given ABCD is a parallelogram. Therefore, 

AB || DC  &  AD || BC

Now, AB || DC and AD is Intersecting them at A and D respectively.

\angle      A + \angle      D = 180\degree                     …(i)             [ Sum of consecutive interior angles is 180\degree      ]

Now, AD || BC and DC is Intersecting them at D and C respectively.

\angle      D + \angle      C = 180\degree                    …(ii)            [ Sum of consecutive interior angles is 180\degree]

From (i) and (ii) , we get

\angle      A + \angle      D = \angle      D  +  \angle      C

So,  \angle      A = \angle      C



Similarly, \angle      B = \angle      D

\angle      A = \angle      C and  \angle      B = \angle      D

Hence Proved !

Proof 3: Diagonals of a parallelogram bisect each other.

Given: ABCD is a parallelogram

To Prove: OA = OC & OB = OD

As given ABCD is a parallelogram. Therefore,

AB || DC  &  AD || BC

Now, AB || DC and AC is intersecting A and C respectively.

\angle      BAC = \angle      DCA                               [ Alternate Interior Angles are equal ]



So, \angle      BAO = \angle      DCO

Now,  AB || DC and BD is intersecting B and D respectively.

\angle      ABD = \angle      CDB                               [ Alternate Interior Angles are equal ]

So, \angle      ABO = \angle      CDO

Now, in  \triangle      AOB &  \triangle      COD we have, 

\angle      BAO = \angle      DCO                               [ Opposite sides of a parallelogram are equal ]

AB = CD

\angle      ABO = \angle      CDO

So, by ASA(Angle-Side-Angle) congruence criterion 

\triangle      AOB  \cong        \triangle      COD

OA = OC and OB = OD

Hence Proved !




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