In this video we shall look at the different properties of a parallelogram.
We have studied that a parallelogram is a quadrilateral in which pair of opposite sides are parallel.
The other properties of a parallelogram are:
1) Each diagonal of a quadrilateral divided it into two congruent triangles.
We have illustrated this through a diagram. Suppose we have a parallelogram ABCD where AD is parallel to BC and AB is parallel to DC. AC and BD are the diagonals of the parallelogram.
Diagonal AC divides parallelogram ABCD into ΔABC and ΔADC whereas diagonal BD divides parallelogram ABCD into ABD and ΔDBC. We take ΔABD and ΔDBC and prove them congruent. We know that AB is parallel to DC, thus diagonal BD acts as a transversal for the parallel lines.
∠ABD will thus be equal to ∠BDC. (Alternate angles)
And ∠ADB = ∠DBC (alternate angles). BD is common to both so BD = BD.
Thus ΔABD is congruent to ΔBDC by the ASA rule.
2) Opposite angles of a parallel are equal.
We have illustrated this through a parallelogram ABCD. In the previous example we saw that by the property of alternate angles, ∠ABD = ∠BDC and ∠ADB = ∠DBC. Therefore ∠ABC = ∠ADC. In this way opposite angles of a parallelogram are equal.
3) Diagonals of a parallelogram bisect each other.
We can illustrate this through an example. suppose we have a parallelogram ABCD where AC and BD are two diagonals that meet at point O. to prove that AO = OC and DO = OB, we will have to prove ΔAOB congruent to ΔOBC.
We know that AB is parallel to DC and AD is parallel to BC. Diagonals AC and BD act as transversals to these parallel lines. Therefore,
∠DAO = ∠BCO (alternate angles)
∠ADO = ∠OBC (alternate angles)
∠ AOD = ∠BOC (vertically opposite angles)
Thus, ΔAOB is congruent to ΔOBC.
When two triangles are congruent, their corresponding sides are in equal ratio. Since AD = BC (opposite sides of a parallelogram are equal). Therefore,
AO = OC and
DO = OB.
Thus, diagonals AC and BD intersect each other.
A rhombus is quadrilateral with all sides equal. It is a type of parallelogram in which a pair of adjacent sides is equal. Since a rhombus is a parallelogram all properties of a parallelogram apply to a rhombus. Further, in a rhombus, the diagonals bisect each other perpendicularly.
Conversely, any quadrilateral where the two diagonals bisect each other at right angles will be a rhombus.
A rectangle is a special type of parallelogram. All properties of a parallelogram apply to a rectangle as well. In addition in a rectangle, each of the angles is equal to 90º. The diagonals of a rectangle are equal and bisect each other.