Properties of Parallelogram: Definition and Formulas

Properties of Parallelogram: A parallelogram is a type of quadrilateral in which the opposite sides are parallel and equal. A parallelogram is a quadrilateral and there are four angles at the vertices. It is imperative that you understand the properties of Parallelogram which will be helpful for calculations in problems relating to the sides and angles of a parallelogram in 2-D Geometry. Apart from that, there are many other attributes of a parallelogram that we will uncover and explain in detail in this article. 

What Is a Parallelogram?

In Euclidean geometry, a parallelogram is a special type of quadrilateral that has equal and parallel opposite sides. The opposite sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

In the figure above, ABCD is a parallelogram. Here, AB || CD and AD || BC. 
Also, AB = CD and AD = BC
∠A = ∠C and ∠B = ∠D

Also, ∠A and ∠D are supplementary angles because these interior angles lie on the same side of the transversal. Similarly, ∠B and ∠C are supplementary angles.

Hence,
∠A + ∠D = 180°
∠B + ∠C = 180°

Now that you are clear on parallelogram definition, let’s see what are all the properties of a parallelogram.

Check properties of other Geometric shapes also:

Properties of Parallelogram

Basically, there are 6 properties of a parallelogram which are important. We will see each property in detail:

1. Opposite sides are parallel and congruent.
2. Opposite angles are congruent.
3. The consecutive angles of a parallelogram are supplementary.
4. If one angle of a parallelogram is right, then all angles are right.
5. The diagonals of a parallelogram bisect each other and each one separates the parallelogram into two congruent triangles.
6. Parallelogram law.

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Let’s understand all the properties of a parallelogram with an example:

Property 1: Opposite sides are equal and congruent.

In the parallelogram ABCD, the first property states that,

AD ≅ BC and CD ≅ AB.
The length of AD is equal to BC and the length of CD is equal to AB. Moreover, they are congruent which means the opposite sides are identical, and they coincide exactly when they are superimposed, or placed on top of each other.

Property 2: Opposite angles are equal and congruent.

The second property states that,

∠A ≅ ∠C and ∠B ≅ ∠D.
Opposite angles are of equal measure and they are congruent to each other.

Property 3: Consecutive angles in a parallelogram are supplementary.

This property of a parallelogram states that the adjacent angles of a parallelogram are supplementary. It means the sum of the two adjacent angles is 180°

Here,
∠A + ∠D = 180°
∠B + ∠C = 180°
∠A + ∠B = 180°
∠C + ∠D = 180°

Please note that the sum of all the interior angles of a parallelogram is 360°.

Property 4: If one angle of a parallelogram is a right angle, then all angles are right angles.

If a parallelogram is known to have one right angle, then with the help of the co-interior angles property, it can be proved that all its angles are right angles.

Hence, such a parallelogram becomes a ‘Rectangle‘.

Proof:

Consider ABCD to be a parallelogram and the measure of angle C is 90°.
Now, since ABCD is a parallelogram, then the measure of angle will be equal to the measure of angle A, that is,
m∠C = m∠A
=90° (Opposite angles of a parallelogram are equal)
and m∠A + m∠B = 180 (Corresponding angles)
⇒ 90 + m∠B = 180
⇒ m∠B = 90
Also, m∠B = m∠D
=90° (Opposite angles of a parallelogram are equal).
Thus, m∠A = m∠B = m∠C = m∠D = 90°.
Since all the angles of the parallelogram are equal to 90°, therefore ABCD is a rectangle, by the properties of a rectangle.
Hence proved.

Property 5: Properties of diagonals of a parallelogram.

There are two attributes of diagonals of parallelogram:

(i) The two diagonals bisect each other.
(ii) Each diagonal bisects the parallelogram into two congruent triangles.

The first attribute states that the two diagonals ‘AC’ and ‘BD’ of parallelogram ‘ABCD’ bisect each other. In other words,
AE = EC and DE = EB.

The second attribute states that diagonal AC divides the parallelogram ABCD into two congruent triangles, △ADC and △ABC. Similarly, diagonal BD divides the parallelogram ABCD into two congruent triangles, △ABD and △BCD.
Hence, △ADC ≅ △ABC
△ABD ≅ △BCD

Property 6: Parallelogram law

The sixth property of parallelogram is also called parallelogram law. This law states that the sum of the square of all the sides of a parallelogram is equal to the sum of the square of its diagonals.

In other words, in a parallelogram ABCD, we have:
AB² + BC² + CD² + AD² = AC² + BD²

Proof:

Let AD = BC = x, AB = CD = y, and ∠ BAD = α
Using the law of cosines in the △BAD, we get
x² + y² – 2xy cos(α) = BD² ——-(1)
We know that in a parallelogram, the adjacent angles are supplementary. So
∠ADC = 180 – α
Now, again use the law of cosines in the △ADC
x² + y² – 2xy cos(180 – α) = AC² ——-(2)
Apply trigonometric identity cos(180 – x) = – cos x in (2)
x² + y² + 2xy cos(α) = AC²
Now, the sum of the squares of the diagonals (BD² + AC²) are represented as,
BD² + AC²  = x² + y² – 2xycos(α) + x² + y² + 2xy cos(α)
Simplify the above expression, we get;
BD² + AC² =2x² + 2 y² ——-(3)
The above equation is represented as:
BD² + AC² = 2(AB)² + 2(BC)²
Hence, the parallelogram law is proved.

Solved Examples On Parallelogram

Here we have provided some parallelogram examples with solutions:

Question 1: Find the measures of ∠C and ∠B in the parallelogram ABCD.

Solution: By the second property of a parallelogram, opposite angles of a parallelogram are congruent. So, we have
m∠C = m∠A
From the diagram, m∠A = 70°.
m∠C = 70°

By the third property of parallelogram, the consecutive angles of a parallelogram are supplementary. So, we have
m∠A + m∠B = 180°
From the diagram, m∠A = 70°.
70° + m∠B = 180°
Subtract 70° from both sides.
m∠B = 110°

Question 2: A parallelogram, with dimensions in cm, is shown below:

What is the perimeter of the parallelogram, in cm?

Solution: The triangle on the left side of the figure has a 30° and a 90° angle. Since all of the angles of a triangle must add up to 180°, we can find the angle measure of the third angle:
30 + 90 = 120
180 − 120 = 60°

Our third angle is 60° and we have a 30−60−90 triangle.
A 30−60−90 triangle has sides that are in the corresponding ratio of x−x√3−2x. In this case, the side opposite our 30° angle is 6, so
x=6
We also now know that
x√3 = 6√3
∴ 2x = 12

Now we know all of our missing side lengths. The right and left sides of the parallelogram will each be 12. The bottom and top will each be 20+6√3. Let’s combine them to find the perimeter:
Perimeter = (2w+2l)
2⋅(12) + 2⋅(20 + 6√3)
24 + 40 + 12√3
64+12√3

Question 3: Find the perimeter of the following parallelogram:

Solution: The formula for the perimeter of a parallelogram is:
P = 2(side) + 2(base).

Putting in our values, we get:
P = 2(4√2m) + 2(19 m)
P = 38 + 8√2m

Question 4: In the parallelogram ABCD, what is the value of x?

Solution: Opposite angles are equal, and adjacent angles must sum to 180°.
Therefore, we can set up an equation to solve for z:
(z – 15) + 2z = 180
3z – 15 = 180
3z = 195
z = 65
Now solve for x:
2z = x
= 130°

Get Maths formulas for Class 6 to 12 below:

Properties of Parallelograms: Practice Questions

Here we have provided some of the practice questions on properties of parallelogram Class 9 for you to practice:

FAQs on Properties of Parallelogram

Here we have provided some of the frequently asked questions related to parallelogram and their properties:

Now you are provided with all the necessary information regarding the properties of parallelograms. Practice more questions and master this concept. Students can make use of NCERT Solutions for Maths provided by Embibe for their exam preparation.

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