Parallelogram

Parallelogram Definition

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length & opposite angles are equal in measures.

Area of parallelogram-

Area = base $\times$ height

A = a$\times$b$\times$ sinA = b $\times$a$\times$sinB

Properties of parallelogram

1. Opposite sides have same length AB = CD, AD = BC.
2. Opposite sides are parallel AB||CD & AD||BC
3. Opposite angles are equal ∠ABC = ∠CDA & ∠BCD = ∠DAB
4. Sum of angles is equal to 360, i.e. ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°
5. The sum of the angles adjacent to any sides is 180.
∠ABC + ∠BCD = ∠BCD + ∠CDA = ∠CDA + ∠DAB = ∠DAB + ∠DAB = 180°
6. Each diagonal divides the parallelogram into two equal triangles.
7. Bisectors of adjacent parallelogram angle always intersect at right angles (90°)
 Example- Find the area of a parallelogram whose base is 5 cm and height is 8 cm. Solution- Given, Base = 5 cm and Height = 8 cm. We know, Area = Base x Height Area = $5 \times 8 \;cm^{2}$ Area = $40 \;cm^{2}$ Example: Find the area of a parallelogram having length of diagonals to be 10 and 22 cm and an intersecting angle to be 65 degrees. Solution: We know that the diagonals of a parallelogram bisect each other, hence the length of half the diagonal will be 5 and 11 cm. The angle opposite to the side b comes out to be 180 – 65 = 115$^{\circ}$ We use the law of cosines to calculate the base of the parallelogram – b² =  5² + 11² – 2(11)(5)cos(115°) b² =  25 + 121 – 110(-.422) b² = 192.48 b = 13.87 cm. After finding the base we need to calculate the height of the given parallelogram. To find the height we have to calculate the value of θ, so we use sine law 5/sin(θ) = b/sin(115) θ = 19.06 Now we extend the base and draw in the height of the figure and denote it as ‘h’. The right-angled triangle (marked with red line) has the Hypotenuse to be 22 cm and Perpendicular to be h. So sin θ = h/22 h = 7.184 cm Area = base $\times$ height A = 13.87 $\times$ 7.184 A = 99.645 $cm ^{2}$

Practise This Question

ABCD is a parallelogram in which DCB and ABC are in ratio 2:7. Find the sum of ABD and ADB.