Quadrilaterals with Congruent Opposite Angles are Parallelograms

In last week’s post, we have learned that quadrilaterals with congruent opposite sides are parallelograms. In this post, we show a related theorem. That is, quadrilaterals whose opposite angles are congruent are parallelograms.

quadrilateral abcd

In the figure above, ABCD is a quadrilateral where \angle A and \angle C are congruent and \angle D and \angle B are congruent. In proving the theorem, we need to show that the opposite sides of ABCD are parallel. That is \overline {AB} is parallel to \overline{CD} and \overline {AD} is parallel to \overline {BC}.

Theorem

If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Given

ABCD with A \cong C and B \cong D.

What To Show

AB \parallel CD and AD \parallel BC.

Proof

Since both ABC and ADC are triangles, by the Triangle Angle Sum Theorem, the sum of their interior angles are equal (both  180^\circ).

Therefore,

m \angle 1 + m \angle 2 + m \angle D = m \angle 3 + m \angle 4 + m \angle B.

Since from the given, m \angle B \cong m \angle D,

m \angle 1 + m \angle 2 = m \angle 3 + m \angle 4. (1)

Also, since \angle A \cong \angle C,

m \angle 1 + m \angle 3 = m \angle 2 + m \angle 4. (2)

Subtracting (2) from (1), we have

m \angle 2 - m \angle 3 = m \angle 3 - m \angle 2

which results to m \angle 2 = m \angle 3.

Therefore \angle 2 \cong \angle 3.

Since \angle 2 and \angle 3 are congruent, by the Parallel Postulate,

AD \parallel BC.

Using the steps similar to the proof above, it can be shown that \overline{AD} \parallel \overline{BC} (left as an exercise).

Therefore, ABCD is a parallelogram.

So, if opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Leave a Reply