# Properties of Parallelograms

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:

- Number of sides in Parallelogram = 4
- Number of vertices in Parallelogram = 4
- Area = Base x Height
- Perimeter = 2 (Sum of adjacent sides length)
- 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 = BCFirstly, 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]

Now, In ADC & CBA

DAC = BCA [ From (i) ]

AC = AC [ Common Side ]

DCA = BAC [ From (ii) ]

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

ADC 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:A = C and B = DAs 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 = ODAs 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 !