1. What is a Parallelogram?
Parallelogram - It is a quadrilateral with opposite sides equal and parallel. Diagonally opposite angles in a parallelogram are also equal. Since a parallelogram is also a quadrilateral, the sum of all the angles of a quadrilateral is 360 degrees.
Quadrilaterals like squares and rectangles also have opposite sides equal and parallel. Diagonally opposite angles are also equal in these shapes(squares and rectangles have all right angles). So Square and rectangle is also a parallelogram.
Rhombus is also a parallelogram as it has opposite sides equal and parallel and opposite angles are also equal.
Other quadrilaterals like trapezium and kite do not satisfy the properties of a parallelogram.
1.1 Properties of parallelogram
Let us look at all the properties of a parallelogram.
- Opposite sides are equal.
Here, AB = CD and BC = AD
- Opposite sides are parallel.
Here, AB is parallel to CD and BC is parallel to AD.
- Opposite angles are equal.
Here, a = c, b = d
- Adjacent angles add up to 180 degrees.
Here, a + b = 180°, b + c = 180°, c + d = 180°, d + a = 180°
2. Working out angles in a parallelogram
We have seen that angles in a parallelogram add up to 360 degrees. Using this and the above mentioned properties on parallelogram, let us try to work out some of the missing angles in quadrilaterals.
2.1 Solved example
ABCD is a parallelogram. If a = 65°, calculate the size of angle x.
Solution: x and a are adjacent angles.
From the property of parallelogram, adjacent angles add up to 180°.
So x + a = 180°
x + 65° = 180°
x = 180° - 65° = 115°
The value of x is 115°.
For the above example question, you can also try to work out the remaining angles of the parallelogram using the properties of a parallelogram.
3. Perimeter and Area of parallelogram
Perimeter of parallelogram: It is the sum of all the four sides of the parallelogram.
For the parallelogram shown below,
Perimeter = a + b + a + b
= 2a + 2b = 2(a+b)
Formula for the perimeter of a parallelogram is similar to the perimeter of a rectangle.
Area of a parallelogram:
Below is the formula to find the area of the parallelogram using its height and base.
Area of a parallelogram = b × h , where b is the base and h is the vertical height of the parallelogram.
3.1 Solved examples
Example 1:
What is the perimeter of the parallelogram shown below, if a = 15 cm, b = 12 cm?
Solution: Perimeter of a parallelogram = 2(a+b)
= 2(15+12)
= 54 cm
Example 2:
The diagram below shows a parallelogram drawn inside a square.
What is the area of the shaded region?
Solution: Area of the shaded region is nothing but the area of the parallelogram.
Base of the parallelogram b = 12 - 2.5 = 9.5 feet
Height of the parallelogram h= Side of the square = 12 feet
Area of a parallelogram = b × h
= 9.5 × 12 = 114 feet²
Area of the shaded region is 114 feet².
