Parallelogram Definition
Understanding quadrilaterals of Class 8
Parallelogram
Properties of parallelogram are as follows:








Area of Parallelogram (A) : Base (b) . Height (h)
Theorem: In a parallelogram, prove that
 the opposite sides are equal;
 the opposite angles are equal;
 diagonals bisect each other.
Proof: Let ABCD be a parallelogram. Draw its diagonal AC.
In triangles ABC and CDA, we have ∠1 = ∠3 [Alternate angles] ∠2 = ∠4 [Alternate angles] and, AC = AC [Common side] So, by ASA congruence criterion, we have ΔABC≅ΔCDA ⇒ AB = CD, BC = AD and ∠B = ∠D 

Corresponding parts of congruent triangle are equal]
Similarly, by drawing the diagonal BD, we can prove that
ΔABD ≅ΔCDB
⇒ AD = BCand∠A = ∠C
Hence, in parallelogram ABCD, we have
AB = CD, AD = BC, ∠A = ∠C, ∠B = ∠D This proves (i) and (ii) In order to prove (iii), consider a parallelogram ABCD. Draw its diagonals AC and BD, intersecting each other at O. In Δs AOB and COD, we have 

AB = CD [Opposite sides of a parallelogram are equal]
∠AOB = ∠COD [Vertically opposite angles]
∠OAB = ∠DCO [Alternate angles]
∴ ΔOAB≅ΔOCD
⇒ OA = OC and OB = OD
This proves (iii).
Types of Parallelogram:
Rectangle
Rectangle is parallelogram with a right angle. It has the following properties.


Rhombus
Rhombus is a parallelogram whose adjacent sides and adjacent angles are equal. So, rhombus has all the properties of a parallelogram.
Hence, from the properties of a parallelogram, we have


Square
A square is an equilateral rectangle. This means a square has all the properties of a rectangle with an additional requirement that all the sides have equal length. In square, the diagonals 

 bisect one another
 are of equal length
 are perpendicular to one another.
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