Properties Of Parallelograms

Learn about the properties of parallelograms including relationships among opposite sides, opposite angles, adjacent angles, diagonals and angles formed by diagonals.

Parallelogram: A quadrilateral with two pairs of parallel lines.

To start off with basic rules, opposite sides of a parallelogram are always equal length and parallel.

Inside a parallelogram, opposite angles are always congruent. Angles that lie next to each other are always supplementary.

The diagonals in a parallelogram bisect each other. When the diagonals are drawn, this creates many angles that follow the same rules as do the angles formed by two parallel lines intersected by a transversal. Alternate interior angles are congruent, corresponding angles are equal, vertical angles are equal. Same side interior angles are supplementary.

Video-Lesson Transcript

Let’s go over parallelograms.

We already drew parallelogram $ABCD$.

It’s called a parallelogram because there are two pairs of parallel sides.

Side $BC$ is parallel to side $AD$. And side $AB$ is parallel to side $CD$.

Besides these two pairs of sides being parallel, they are also congruent.

$\overline{\rm BC} \cong \overline{\rm AD}$
$\overline{\rm AB} \cong \overline{\rm CD}$

Angles across from each other are congruent.

$m\angle A \cong m\angle C$
$m\angle B \cong m\angle D$

The other way to pair the angles is to make them supplementary.

Angles that are adjacent or next to each other add up to $180$.

$m\angle A + m\angle D = 180$
$m\angle A + m\angle B = 180$
$m\angle C + m\angle B = 180$
$m\angle C + m\angle D = 180$

Angles across from each other are congruent. And if they are next to eaach other, they add up to $180$.

Let’s see what happens when we draw diagonals.

Let’s draw a diagonal from $A$ to $C$ and from $B$ to $D$.

Diagonals actually bisect each other.

$BD$ splits $AC$ in half.

Likewise, $AC$ splits $BD$ in half.

Something else that happens when we draw this diagonal is it affects the angles.

Let me extend this diagonal line and these lines.

Let’s look at these red lines. We’ll see that there are parallel lines and they are intersected by a transversal.

And we know that there is a relationship that happens with the angles.

Because we have a lot of parallel lines intersected by a transversal.

These are all ultimate interior angles.