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