# Properties of Parallelograms

Last Updated : 22 Apr, 2024

Parallelogram is a quadrilateral in which opposite sides are parallel and congruent and the opposite angles are equal. A parallelogram is formed by the intersection of two pairs of parallel lines.

## Important Properties of a Parallelogram

A parallelogram is a quadrilateral in which opposite sides are parallel and equal in length. Also, opposite angles are also equal. Few of the important properties of a parallelogram are:

• Opposite sides of a parallelogram are equal in length and are parallel to each other.
• Opposite angles in a parallelogram are equal.
• Sum of all interior angles of a parallelogram is 360Â°.
• Consecutive angles of a parallelogram are supplementary (180Â°).

### Properties of Parallelogram Diagonals

Various properties of the diagonal of a parallelogram are:

• Both diagonals of a parallelogram bisect each other.
• The parallelogram is bisected into two congruent triangles by each diagonal.
• By parallelogram law, Sum of squares of the diagonal of a parallelogram is equal to the sum of squares of all the sides of a parallelogram.

## Theorems on Properties of a Parallelogram

Few important theorems on properties of a Parallelogram are:

• Opposite sides of a parallelogram are equal.
• If opposite sides are equal in a quadrilateral, then it is a parallelogram
• Opposite angles of a parallelogram are equal.
• If the opposite angles in a quadrilateral are equal, then it is a parallelogram
• Diagonals of a parallelogram bisect each other.
• If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Let’s discuss these theorems in detail with proof.

### Theorem 1: Opposite sides of a parallelogram are equal.

Given: ABCD is a parallelogram

To Prove: AB = CD & DA = BC

Â

Proof:

Given ABCD is a parallelogram. Therefore,Â

AB || DC Â & Â AD || BC

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

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

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

âˆ BAC =Â âˆ D …(ii) Â  Â  Â  Â  Â  Â  Â  Â  Â  Â [Alternate Interior Angles]

âˆ DAC =Â âˆ BCA [ From (i) ]
AC = AC [ Common Side ]
âˆ DCA =Â âˆ 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

### Converse of Theorem 1: If opposite sides are equal in a quadrilateral, then it is a parallelogram

Given: Opposite sides in a quadrilateral ABCD are equal, AB = CD, and BC = AD.

To Prove: Quadrilateral ABCD is a parallelogram.

In quadrilateral ABCD, AB = CD and AD = BC. In triangles ABC, and CDA we have

AC = AC (Common sides)
AB = CD (since alternate interior angles are equal)

So by the SSS congruency criterion, triangles ABC, and CDA are congruent, thus by CPCT corresponding angles of triangles are equal. Thus, âˆ BAC = âˆ DCA, and âˆ BCA = âˆ DAC.

Now AB || CD, BC || AD and thus ABCD is a parallelogram.

### Theorem 2: Opposite angles of a parallelogram are equal.

Â

Given: ABCD is a parallelogram

To Prove:Â Â âˆ A =Â âˆ C Â andÂ âˆ B =Â âˆ D

Proof:

Given ABCD is a parallelogram. Therefore,Â

AB || DC Â & Â AD || BC

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

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

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

âˆ D +Â âˆ C = 180Âº Â  Â  Â  Â  Â  Â  Â …(ii) Â  Â  Â  Â  Â  Â [ Sum of consecutive interior angles is 180Âº]

From (i) and (ii) , we get

âˆ A +Â âˆ D =Â âˆ D Â + Â âˆ C

So, Â âˆ A =Â âˆ C

Similarly,Â âˆ B =Â âˆ D

âˆ A =Â âˆ C and Â âˆ B =Â âˆ D

Hence Proved

### Converse of Theorem 2: If the opposite angles in a quadrilateral are equal, then it is a parallelogram

Given: In the quadrilateral ABCD âˆ A = âˆ C and âˆ B = âˆ DÂ

To Prove: ABCD is a parallelogram.

Proof:

Given âˆ A = âˆ C and âˆ B = âˆ D in quadrilateral ABCD. We have to prove ABCD is a parallelogram

âˆ A + âˆ B + âˆ C + âˆ D = 360Âº (given âˆ A = âˆ C and âˆ B = âˆ D )

2(âˆ A + âˆ B) =360Âº

âˆ A + âˆ B = 180Âº.

Thus AD || BC. Similarly, we can show that AB || CD.Â

Hence, AD || BC, and AB || CD. Therefore ABCD is a parallelogram.

### Theorem 3: Diagonals of a parallelogram bisect each other.

Given: ABCD is a parallelogram

To Prove: OA = OC & OB = OD

Proof:

AB || DC Â & Â AD || BC

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

âˆ BAC =Â âˆ DCA Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  [ Alternate Interior Angles are equal ]

So,Â âˆ BAO =Â âˆ DCO

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

âˆ ABD =Â âˆ CDB Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  [ Alternate Interior Angles are equal ]

So,Â âˆ ABO =Â âˆ CDO

Now, in Â Î”AOB & Â Î”COD we have,Â

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

AB = CD

âˆ ABO =Â âˆ CDO

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

Î”AOB â‰… Â Î”COD

OA = OC and OB = OD

Hence Proved

### Converse of Theorem 3: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Given: The diagonals AC and BD bisect each other.

To Prove: ABCD is a parallelogram.

Proof:

If the diagonals AB and CD bisect each other. Then in Î” AOB, and Î” COD

AO = CO (Diagonals bisect each other)
BO = DO (Diagonals bisect each other)
âˆ AOB =âˆ COD (vertically opposite angles)

Thus, by SAS congruency criterion, triangles are congruent. So Â âˆ CAB = âˆ DCA, and âˆ DBA = âˆ CDB. Hence, AB || CD, and BC || AD. Thus ABCD is a parallelogram.

## Conclusion

In conclusion, parallelograms are fundamental part of geometry. Their unique properties, such as congruent opposite sides and angles, and diagonals that bisect each other, make them applicable in various fields, from architecture to engineering. Understanding these properties enhances our ability to solve geometric problems and appreciate the mathematical harmony in the world around us.

Related Resources

## Solved Example on Properties of Parallelogram

Example 1: ABCD is a quadrilateral with AB = 10 cm. Diagonals of ABCD bisect each other at right angles. Then find the perimeter of ABCD.

Solution:

We know that, if diagonals of a quadrilateral bisect each other at right angles then it is a rhombus.

Thus, ABCD is a rhombus and Â AB = BC = CD = DA.

Thus, the perimeter of ABCD = 4(AB) = 4(10) = 40 cm

Example 2: Find area of a parallelogram where the base is 6 cm and the height is 12 cm.

Solution:

Given, Base = 6 cm and Height = 12 cm.

We know,

Area = Base x Height

Area = 6 Â Ã— 12

Area = 72 cm2

## FAQs on Properties of a Parallelogram

### What is a parallelogram?

A parallelogram is a quadrilateral that has in which opposite sides are parallel and equal. And opposite angles in a parallelogram equal. It is also considered as a cyclic quadrilateral.

### Can a Rhombus be called a Parallelogram?

Opposite sides of a rhombus are equal and parallel, and its opposite angles are also equal. So it is considered a parallelogram.

### What are the four important properties of a parallelogram?

The four important properties of the parallelogram are:

• Opposite sides of a parallelogram are parallel and congruent
• Consecutive angles of a parallelogram add up to 180 degrees
• Opposite angles of a parallelogram are equal
• Diagonals of a parallelogram bisect each other.

### What is the order of rotational symmetry in a parallelogram?

Order of rotational symmetry is a parallelogram is 2.

### Does a parallelogram have reflectional symmetry?

No, reflectional symmetry is not possible in a parallelogram.

### Are the Diagonals of a Parallelogram Equal?

Diagonals of a parallelogram are generally NOT equal. In some special cases, parallelograms such as squares and rectangles have equal diagonals.

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