Quadrilaterals With Congruent Opposite Angles Are Parallelogram

Posted on 19 October, 2014 by Math Proofs Leave a comment

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).

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