# Quadrilaterals With Congruent Opposite Angles Are Parallelogram

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 $\angle A \cong \angle C$ and $\angle B \cong \angle D$. To show that it is a parallelogram, we have to show that $AB \parallel CD$ and $AD \parallel BC$.

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

It is given that $\angle A \cong \angle C$ and $\angle B \cong \angle D$

Since the sum of the interior angles of a quadrilateral is 360 degrees, we have $m \angle A + m \angle B + m \angle C + m \angle D = 360$

We substitute $\angle C$ with $\angle A$ and $\angle B$ with $\angle D$ since these pairs are congruent opposite angles. $m \angle A + m \angle D + m \angle A + m \angle D = 360$ $2 (m \angle A) + 2(m \angle D) = 360$

Dividing both sides by 2, we have $m \angle A + m \angle D = 180$. (1)

On the other hand, if we extend DA to form angle DAE as shown, then we form $\angle BAE$. Now, $\angle BAE + \angle BAD = 180$ (2)

which means that $\angle BAE = \angle D$ from (1) and (2).

But $\angle BAE$ and $\angle D$ are corresponding angles. Therefore, $AB \parallel CD$ which is what we want to show.

It can be easily shown using a similar argument, $AD \parallel BC$ (left as an exercise).