To explore when a quadrilateral is a parallelogram, we will consider some different quadrilaterals.

Case $1$

For the first case, let's consider a quadrilateral whose opposite angles are congruent.

By the Corollary to the Polygon Interior Angles Theorem, the sum of the measures of the interior angles of

A B C D

is

36 0^{\circ} .

x^{\circ} + y^{\circ} + x^{\circ} + y^{\circ} = 36 0^{\circ} ⇓ 2 (x^{\circ} + y^{\circ}) = 36 0^{\circ} ⇓ x^{\circ} + y^{\circ} = 18 0^{\circ}

The latter equation implies that both

∠ A

and

∠ B,

as well as

∠ B

and

∠ C,

are supplementary. Then, by the Consecutive Interior Angles Converse we have that

\overline{A D} ∥ \overline{B C}

and

\overline{A B} ∥ \overline{C D},

which proves that

A B C D

is a parallelogram.

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

Case $2$

For the second case, let's consider a quadrilateral with both pairs of opposite sides congruent.

If we draw $\overline{A C},$ it will divide $A B C D$ into two triangles, and by the Side-Side-Side (SSS) Congruence Theorem, the two triangles are congruent, which implies that $∠ B ≅ ∠ D .$

Drawing $\overline{B D}$ and using a similar reasoning we conclude that $∠ A ≅ ∠ C .$

Applying what we learned in case $1,$ we conclude that $A B C D$ is a parallelogram. Thus, we can write the following statement.

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Case $3$

Finally, let's consider a quadrilateral such that its diagonals bisect each other. Notice that in this case, the quadrilateral is divided into four triangles.

By the Vertical Angles Congruence Theorem, we have that $∠ C M B ≅ ∠ A M D$ and $∠ B M A ≅ ∠ D M C .$

Next, by applying the Side-Angle-Side (SAS) Congruence Theorem, we obtain that $△ A M D ≅ △ C M B$ and $△ A M B ≅ △ C M D .$ These two congruences imply that $\overline{A D} ≅ \overline{B C}$ and $\overline{A B} ≅ \overline{C D} .$ Thus, by case $2,$ we conclude that $A B C D$ is a parallelogram.

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Case 1

Case 2

Case 3

Case $1$

Case $2$

Case $3$

Exercise