In the **previous post**, we have proved that if the opposite sides of a quadrilateral are congruent, then they are parallel. In this post, we are going to show that if the opposite angles of a quadrilateral are congruent, then it is a parallelogram.

In the figure below, we have quadrilateral ABCD with and . To show that it is a parallelogram, we have to show that and .

**Theorem:** If two pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram.

**Proof**

It is given that and .

Since the sum of the interior angles of a quadrilateral is 360 degrees, we have

We substitute with and with since these pairs are congruent opposite angles.

Dividing both sides by 2, we have

. (1)

On the other hand, if we extend DA to form angle DAE as shown, then we form .

Now,

(2)

which means that

from (1) and (2).

But and are corresponding angles. Therefore,

which is what we want to show.

It can be easily shown using a similar argument, (left as an exercise).