CBSE Class 8 Mathematics – 16. Parallelograms (Notes)

Revision Notes on Parallelograms

Quadrilateral

Any closed polygon with four sides, four angles and four vertices are called Quadrilateral.

It could be a regular or irregular polygon.

Angle sum property of a Quadrilateral

Sum of all the interior angles of a Quadrilateral = 360°

Sum of all the exterior angles of a Quadrilateral = 360°

Kinds of Quadrilaterals

There are different types of the quadrilateral on the basis of their nature of sides and their angle.

1. Trapezium

If a quadrilateral has one pair of parallel sides then it is a Trapezium.

Here AD∥BC in quadrilateral ABCD, hence it is a trapezium.

Remark: If the non-parallel sides of a trapezium are equal then it is called Isosceles Trapezium.

2. Kite

If the two pairs of adjacent sides are equal in a quadrilateral then it is called a Kite.

Here AB = BC and AD = CD

Properties of a kite

The two diagonals are perpendicular to each other.

One of the diagonal bisects the other one.

∠A = ∠C but ∠B ≠∠D

3. Parallelogram

If the two pairs of opposite sides are parallel in a quadrilateral then it is called a Parallelogram.

Here, AB ∥ DC and BC ∥ AD, hence ABCD is a parallelogram.

Elements of a Parallelogram

Some terms related to a parallelogram ABCD

1. Opposite Sides – Pair of opposite sides are

AB and DC,

AD and BC  

2. Opposite Angles – Pair of opposite angles are

∠ A and ∠C

∠B and ∠D 

3. Adjacent Sides – Pair of adjacent sides are

AB and BC

BC and DC

DC and AD

AB and AD

4. Adjacent Angles – Pair of adjacent angles are

∠A and ∠B

∠B and ∠C

∠C and ∠D

∠A and ∠D

Properties of a Parallelogram

1. The opposite sides of a parallelogram will always be equal.  

Here, AB = DC and AD = BC.

2. The opposite angles of a parallelogram will always be of equal measure.

As in the above figure, ∠A = ∠C and ∠D = ∠B.

3. The two diagonals of a parallelogram bisect each other.

Here in ABCD, AC and BD bisect each other at point E. So that AE = EC and DE= EB.

4. The pair of adjacent angles in a parallelogram will always be a supplementary angle.

Example

If the opposite angles of a parallelogram are (3x + 5) ° and (61– x) °, then calculate all the four angles of the parallelogram.

Solution

As we know that the opposite angles are equal in a parallelogram so

(3x + 5)° = (61 – x) °

3x + x = 61– 5

4x = 56

x = 14°

Now substitute the value of x in the given angles.

(3x + 5)° = 3(14) + 5

= 42 + 5 = 47°

(61 – x)° = 61 – 14

= 47°

Both the angles are 47° as the opposite angles are equal.

Now to find the other angles let one of the adjacent angles to the above angle is z.

47° + z = 180° (adjacent angles are supplementary angles).

Z = 180° – 47°

 = 133°

The fourth angle will also be 133° as the opposite angles are equal.

Hence the four angles of the given parallelogram are 47°, 47°, 133° and 133°.

Some Special Parallelograms

1. Rhombus

If a parallelogram has all the four sides equal then it is called a Rhombus.

Here, AB = BC = CD = AD.

Properties of a Rhombus

A rhombus has all the properties of a parallelogram and a kite.

The special property is that its diagonals are perpendicular bisector to each other.

Hence, AE = EC and DE = EB.

It makes an angle of 90° at the point of intersection of the diagonals.

2. Rectangle

A parallelogram with equal angles and equal opposite sides is called Rectangle.

Here, AB = DC and AD = BC.

Properties of a Rectangle

Opposite sides are parallel and equal.

Opposite angles are equal.

Adjacent angles make a pair of supplementary angles.

Diagonals are of equal length.

Diagonals bisect each other.

All the four angles are of 90°.

3. Square

A rectangle with all the four equal sides is called a Square.

Here, AB = BC = CD = AD.

Properties of a Square

All four sides are equal.

Opposite sides are parallel.

Diagonals are of equal length.

Diagonals are perpendicular bisector to each other.

All the four angles are of 90°.