Let us perform an activity.

Cut out a parallelogram from a sheet of paper and cut it along a diagonal (see Fig. 8.7). 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.

Let us now prove this result.

Theorem 8.1

A diagonal of a parallelogram divides it into two congruent  triangles.

Proof

Let ABCD be a parallelogram and AC be a diagonal (see Fig. 8.8). Observe that the diagonal AC divides parallelogram ABCD into two triangles, namely, Δ ABC and Δ CDA. We need to prove that these triangles are congruent.

 or, diagonal AC divides parallelogram ABCD into two congruent triangles ABC and CDA.

Now, measure the opposite sides of parallelogram ABCD. What do you observe?

You will find that AB = DC and AD = BC.

This is another property of a parallelogram stated below:

Theorem 8.2

In a parallelogram, opposite sides are equal.

You have already proved that a diagonal divides the parallelogram into two congruent triangles; so what can you say about the corresponding parts say, the corresponding sides? They are equal.

So, AB = DC and AD = BC

Now what is the converse of this result? You already know that whatever is given in a theorem, the same is to be proved in the converse and whatever is proved in the theorem it is given in the converse. Thus, Theorem 8.2 can be stated as given below :

If a quadrilateral is a parallelogram, then each pair of its opposite sides is equal. So its converse is :

Theorem 8.3

If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

Can you reason out why?

Let sides AB and CD of the quadrilateral ABCD be equal and also AD = BC (see Fig. 8.9). Draw diagonal AC.

Can you now say that ABCD is a parallelogram? Why?

You have just seen that in a parallelogram each pair of opposite sides is equal and conversely if each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram. Can we conclude the same result for the pairs of opposite angles?

Draw a parallelogram and measure its angles. What do you observe?

Each pair of opposite angles is equal.

Repeat this with some more parallelograms. We arrive at yet another result as given below.

Theorem 8.4

In a parallelogram, opposite angles are equal.

Now, is the converse of this result also true? Yes. Using the angle sum property of a quadrilateral and the results of parallel lines intersected by a transversal, we can see that the converse is also true. So, we have the following theorem :

Theorem 8.5

If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.

There is yet another property of a parallelogram. Let us study the same. Draw a parallelogram ABCD and draw both its diagonals intersecting at the point O (see Fig. 8.10).
 

Measure the lengths of OA, OB, OC and OD.

What do you observe? You will observe that

OA = OC and OB = OD.

or, O is the mid-point of both the diagonals.

Repeat this activity with some more parallelograms.

Each time you will find that O is the mid-point of both the diagonals.

So, we have the following theorem :

Theorem 8.6

The diagonals of a parallelogram bisect each other.

Now, what would happen, if in a quadrilateral the diagonals bisect each other? Will it be a parallelogram? Indeed this is true.

This result is the converse of the result of Theorem 8.6. It is given below:

Theorem 8.7

If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

You can reason out this result as follows:

Note that in Fig. 8.11, it is given that OA = OC and OB = OD.

Example 1

Show that each angle of a rectangle is a right angle.

Solution

Let us recall what a rectangle is.

A rectangle is a parallelogram in which one angle is a right angle.

Let  be a rectangle in which

We have to show that

We have,  and is a transversal (see Fig. 8.12).

(Opposite angles of the parallellogram)

Therefore, each of the angles of a rectangle is a right angle.

Example 2

Show that the diagonals of a rhombus are perpendicular to each other.

Solution

Consider the rhombus ABCD (see Fig. 8.13).

You know that   (Why?)

Now, in and ,

                  (Diagonals of a parallelogram bisect each other)

        (Common)

Therefore,

(SSS congruence rule)

This gives,

But ,     (Linear pair)

So,

or,

So, the diagonals of a rhombus are perpendicular to each other.

Example 3

is an isosceles triangle in which bisects exterior angle and (see Fig. 8.14). Show that

Solution

(ii) Now, these equal angles form a pair of alternate angles when line segments and are intersected by a transversal 

Now, both pairs of opposite sides of quadrilateral are parallel.So, is a parallelogram.

Example 4

Two parallel lines  and  are intersected by a transversal (see Fig. 8.15). Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.

Solution

It is given that and transversal intersects them at points and respectively.

The bisectors of and intersect at and bisectors of and intersect at.

We are to show that quadrilateral is a rectangle.

Now,

(Alternate angles as and p is a transversal).

So, 

ie.,

These form a pair of alternate angles for lines and with as transversal and they are equal also.

So, 

Similarly    

Therefore, quadrilateral is a parallelogram.

Also,     (Linear pair)

So,

or,

or, 

So, is a parallelogram in which one angle is .

Therefore, is a rectangle.

Example 5

Show that the bisectors of angles of a parallelogram form a rectangle.

Solution

Let and S be the points of intersection of the bisectors of and and and , and and respectively of parallelogram (see Fig. 8.16).

 In, what do you observe?

Since bisects and bisects therefore,

Similarly, it can be shown that or (as it was shown for ). Similarly, and .

So, is a quadrilateral in which all angles are right angles.

Can we conclude that it is a rectangle? Let us examine. We have shown that and . So both pairs of opposite angles are equal.

Therefore, is a parallelogram in which one angle (in fact all angles) is and so, is a rectangle.

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:

Theorem 8.8

A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

 Look at Fig 8.17 in which and . Let us draw a diagonal . You can show that by congruence rule.

So, (Why?)

Let us now take an example to apply this property of a parallelogram.

 Example 6

is a parallelogram in which and are mid-points of opposite sides and (see Fig. 8.18). If intersects at and intersects at, show that:

is a parallelogram.

is a parallelogram.

is a parallelogram.

Solution

In quadrilateral ,

Therefore, is a parallelogram                              [From (1) and (2) and Theorem 8.8]

Similarly, quadrilateral is a parallelogram, because

In quadrilateral ,

Similarly, 

So, is a parallelogram.