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.

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.

parallelogram proof

 

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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