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).