Definition of Parallelogram

Definition of parallelogram

A parallelogram is a quadrilateral in which both sides are parallel and equal.

PROPERTIES OF A PARALLELOGRAM

1.The opposite sides of a parallelogram are parallel to each other. So, AB || CD and AD || BC

2.The opposite sides of a parallelogram are equal to each other.

So, AB = CD and AD = BC

3.The opposite angles of a parallelogram are equal.

So, ∠A = ∠C and ∠B = ∠D

4.Adjacent angles of a parallelogram are supplementary (sum of the angles is 180°)

So, ∠C+ ∠D = 180°, ∠B + ∠C = 180°, ∠A + ∠B = 180°, and ∠A + ∠D = 180°

5.Diagonals of a parallelogram bisect each other. 

So, DE = EB and AE = AC

6. Diagonals of a parallelogram divide the parallelogram into two    congruent triangles,

     So, ∆ABC ≅ ∆CDA and ∆ADB ≅ ∆DCB and

7. Sum of the interior angles of a parallelogram is 360 degrees.

SPECIAL TYPES OF PARALLELOGRAMS

RECTANGLE: A rectangle is a parallelogram in which each angle measures 90 degrees.

Properties of a rectangle

1)The opposite sides are equal and parallel

2) All angles are equal and measures 90 degrees.

3)Diagonals are equal

4) Diagonals bisect each other.

Properties of a square

SQUARE: A square is a parallelogram in which all sides are equal and all angle measures 90 degrees.

1) All sides are equal

2)The opposite sides are parallel

3) All angles are equal and measures 90 degrees.

4)Diagonals are equal

5) Diagonals bisect each other.

6) Diagonals are perpendicular to each other.

7) Diagonals bisect the angles at the vertices.

Properties of a Rhombus

1)All sides are equal

2)The opposite sides are parallel

3) The opposite angles are equal

4)The adjacent angles of the rhombus are supplementary.

5) Diagonals bisect each other.

6) Diagonals are perpendicular to each other.

7) Diagonals bisect the angles at the vertices.

AREA OF A PARALLELOGRAM

There are three formulas to calculate the area of a parallelogram 

FIRST METHOD

USING BASE & HEIGHT

The area of the Parallelogram is the product of its base(b) and height(h)

Area = base x height

The height of a parallelogram is the perpendicular distance from the base to the opposite side.

Unit of the area of a parallelogram is in square units like square meters, square centimeters, etc.

SECOND METHOD

USING THE TWO DIAGONALS AND THE INTERSECTING ANGLE 

The area of the parallelogram can also be calculated using the two diagonals and their intersecting angles. The formula is as follows – 

Area =1/2 x d1 x d2 sin (x)

Here, d1 and d2 are the diagonals of the parallelogram and x is the measure of the angle formed where the diagonals bisect.

EXAMPLE

Q1. A parallelogram has diagonals with lengths of 10 m and 9 m. If the angle of intersection of the diagonals is 30°, what is their area?

SOLUTION: Given, Diagonal , d1 = 10m

                               Diagonal, d2 = 9m

And angle of intersection of the diagonals x is 30°

So, Area =1/2 x d1 x d2 sin (x)

               = ½ x 10 x 9 x sin30°

               = ½ x 10 x 9 x ½

               = 5 x 4.5

               = 22.5m

So, The area of the parallelogram is 22.5m2.

THIRD METHOD

USING THE LENGTH OF THE PARALLEL SIDES OF THE PARALLELOGRAM & THE ANGLE BETWEEN THEM.

Another Method of calculating the area of a parallelogram is by using the length of the parallel sides & the angle between them. The formula for the same is as follows –

Area = ab sin(x)

Here, a and b are the length of the parallel sides of the parallelogram and x is the angle between sides “a” and “b” of the parallelogram.

EXAMPLES

Q1. Find the area of a parallelogram if its two parallel sides are 60m and 30m and the angle between them is 30 degrees.

Solution: Let a = 60m and b = 30m

The angle between a and b is 30 degrees.

Area of parallelogram = absinΘ

After substituting the values, we get

A = 60 x 30sin30°

   = 1800 x ½

   = 900 cm2

So, area of parallelogram is 900sq.cm

Q2. The angle between any two sides of a parallelogram is 90 degrees. If the length of the two parallel sides is 8 units and 10 units respectively, then find the area.

Solution: Let a = 8 units and b = 10 units
θ = 90 degrees

Using area of parallelogram formula,
Area = ab sin (θ)
⇒ A = 8 × 10 sin (90°)
⇒ A = 80 sin 90°
⇒ A = 80 × 1 = 80 sq. Units

 PRACTICE QUESTIONS AND SOLUTIONS

Q1. The adjacent sides of a parallelogram are 15m and 6m If the distance between the longer sides is 4m, find the distance between the shorter sides.

SOLUTION: 

Let ABCD be a parallelogram with side DC=15m and corresponding altitude AE = 4m

The adjacent side AD = 6m and the corresponding altitude is CF.

We know that area of a parallelogram is base x height

So, equating them we get

AD x CF = DC x AE

6 x CF = 15 x 4

So CF = 60 ÷ 6 = 10 cm

Hence the distance between the shorter side is 10 cm

Q2. In the parallelogram ABCD the measure of angle A = 82°. Find the measure of all the angles

SOLUTION: Given

In the parallelogram ABCD, ∠A = 82°

Sum of the adjacent Angles of Parallelogram is 180°

Therefore, ∠A + ∠D = 180°

⇒ 82° + ∠D = 180°

⇒ ∠D = 180° - 82°

⇒ ∠D = 98°

As opposite angles of a parallelogram are equal 

∠D = ∠B = 98° and ∠A = ∠C = 82°

Angles of the parallelogram ABCD are ∠A = 82° , ∠B = 98° , ∠C = 82° , ∠D = 98°

Q3. The ratio between two adjacent angles of a parallelogram ABCD are in the ratio of 2: 3. Find the measure of each angle of the parallelogram.

SOLUTION: Let the adjacent angles of a parallelogram ABCD be ∠A and ∠B

Let ∠A = 2x° , ∠B = 3x°

As sum of adjacent angles of a parallelogram is 180°, 

so ∠A + ∠B = 180°

Now 2x + 3x = 180°

5x = 180°

x = 180° / 5

x = 36°

So ∠A = 2x , Hence 2 x 36° = 72°

∠B = 3x , Hence 3 x 36° = 108°

Opposite angles of a parallelogram are equal.

Therefore, ∠A = ∠C = 72°

And ∠B = ∠D = 108°

Q4. The sum of two opposite angles of a parallelogram is 140°. Find the measure of each of its angles.

SOLUTION:

Let ABCD be a parallelogram and let the sum of its opposite angles be 140°

∠A + ∠C = 140°

Opposite angles of a parallelogram are equal.

Therefore,  ∠A = ∠C = x°

⇒ x + x= 140

⇒ 2x = 140

⇒ x = 70°

Therefore, ∠A = ∠C = 70°

As sum of adjacent angles of a parallelogram is 180°, 

so ∠A + ∠B = 180°

⇒ 70 + ∠B = 180°

⇒ ∠B = 180° – 70°

⇒ ∠B = 110°

So, ∠B = ∠D = 110° (opposite angles of a parallelogram are equal)

Q5. The perimeter of a parallelogram is 220 cm. If one of the sides is longer than the other by 20 cm, find the length of each of its sides.

SOLUTION:

Let the lengths of two sides of the parallelogram be x cm and (x+20) cm, respectively.

Then, its perimeter 

=2[x+(x+20)] cm

 =2[x+x+20] cm

  =2[2x+20] cm

= 4x+40 cm

4x+40 =220

⇒4x = 220−40 

⇒4x=180

⇒x=180/4

⇒x=45

Length of one side=45 cm 

Length of the other side⇒ (45 + 20) cm=65 cm

Q6. The lengths of the diagonals of a rhombus are 16 cm and 12 cm respectively. Find the length of each of its sides.

SOLUTION:

Let ABCD be a rhombus.

Let AC and BD be the diagonals of the rhombus intersecting at a point O.

Let AC = 16 cm BD = 12 cm

The diagonals of a rhombus bisect each other at right angles

∴ AO= ½ AC

= ½ x 16

= 8cm

And, BO = ½ BD

= ½ x 12

= 6cm

From the right ΔAOB

AB2 = AO2 + BO2

=  82    +  62

 = 64 + 36

= 100

So, AB = √100cm

= 10cm
Hence, the length of the side AB is 10 cm.

AB = BC = CD = DA = 10 cm (All sides of a rhombus are equal).