# 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.

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

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

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

DAC = BCA                    [ From (i) ]

AC = AC                                   [ Common Side ]

DCA = BAC                 [ From (ii) ]

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

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:  A = C  and B = 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.

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

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

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

From (i) and (ii) , we get

A + D = D  +  C

So,  A = C

Similarly, B = D

A = C and  B = 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.

BAC = DCA                               [ Alternate Interior Angles are equal ]

So, BAO = DCO

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

ABD = CDB                               [ Alternate Interior Angles are equal ]

So, ABO = CDO

Now, in  AOB &  COD we have,

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

AB = CD

ABO = CDO

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

AOB    COD

OA = OC and OB = OD

Hence Proved !

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