Properties of a Parallelogram

Properties of a Parallelogram

Let us perform an activity.
Cut out a parallelogram from a sheet of paper and cut it along a diagonal (see Fig). You obtain two triangles. What can you say about these triangles?
Place one triangle over the other. Turn one around, if necessary. What do you observe?
Observe that the two triangles are congruent to each other.
Repeat this activity with some more parallelograms. Each time you will observe that each diagonal divides the parallelogram into two congruent triangles.

Another Condition for a Quadrilateral to be a Parallelogram
You have studied many properties of a parallelogram in this chapter and you have also verified that if in a quadrilateral any one of those properties is satisfied, then it becomes a parallelogram.
We now study yet another condition which is the least required condition for a quadrilateral to be a parallelogram.
It is stated in the form of a theorem as given below:

The Mid –point Theorem
You have studied many properties of a triangle as well as a quadrilateral. Now let us study yet another result which is related to the mid-point of sides of a triangle. Perform the following activity.

Draw a triangle and mark the mid-points E and F of two sides of the triangle. Join the points E and F (see Fig).
Measure EF and BC. Measure ∠AEF and ∠ABC.
What do you observe? You will find that:

 So,   EF || BC
Repeat this activity with some more triangles.
So, you arrive at the following theorem: