Properties of Parallelograms

Q: If one angle of a parallelogram is $90^\circ$, what are the others?

All $90^\circ$. The opposite angle matches it, and each consecutive angle must be $180 - 90 = 90^\circ$. A parallelogram with one right angle is automatically a rectangle.

A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. That one definition forces a whole package of other facts to be true — about the sides, the angles, and the diagonals — and those facts are what every parallelogram problem runs on.

There are four properties to know. Once you can match the right property to what the problem gives you, most parallelogram questions turn into a one- or two-step equation.

The four properties

In any parallelogram: opposite sides are congruent and parallel, opposite angles are congruent, consecutive angles are supplementary, and the diagonals bisect each other.

Congruent means equal in measure — if $AB = 9$ in parallelogram $ABCD$ , then the opposite side $CD = 9$ too. Supplementary means the two angles add to $180^\circ$ . Bisect means cut exactly in half: the diagonals cross at a point that is the midpoint of both of them.

One warning up front: the diagonals of a general parallelogram are not congruent to each other, and they are not perpendicular. Those are special extras that only show up in rectangles and rhombi.

In the figure, the chevrons mark the two pairs of parallel sides, and the tick marks show the diagonals bisecting each other at $E$ .

Why consecutive angles add to $180^\circ$

Take two angles that share a side, like $\angle A$ and $\angle B$ along side $\overline{AB}$ . The other two sides through those vertices, $\overline{AD}$ and $\overline{BC}$ , are parallel, and $\overline{AB}$ is a transversal cutting across them. Co-interior angles between parallel lines are supplementary, so $m\angle A + m\angle B = 180^\circ$ .

This is the property students forget most. Opposite angles match; neighboring angles add to $180^\circ$ . So if one angle of a parallelogram is $70^\circ$ , the angle across from it is $70^\circ$ and both of its neighbors are $110^\circ$ .

Matching the property to the problem

Read what the problem gives you. Side lengths or expressions on opposite sides — set them equal, because opposite sides are congruent. Two angles across from each other — set them equal. Two angles that share a side — set their sum equal to $180$ . Pieces of the diagonals meeting at the intersection point — set the two halves of the same diagonal equal, because the diagonals bisect each other.

Worked examples

Example 1: finding angles

In parallelogram $ABCD$ , $m\angle A = 70^\circ$ . Find $m\angle C$ and $m\angle B$ .

\angle C

is opposite

\angle A

, and opposite angles are congruent

m\angle C = 70^\circ

\angle B

is consecutive with

\angle A

, and consecutive angles are supplementary

m\angle A + m\angle B = 180^\circ

Solve

m\angle B = 180 - 70 = 110^\circ

Answer: $m\angle C = 70^\circ$ and $m\angle B = 110^\circ$

Example 2: the diagonals bisect each other

In parallelogram $ABCD$ , the diagonals intersect at $E$ . If $AE = 2x + 3$ and $EC = 4x - 7$ , find $AE$ .

E

is the midpoint of diagonal

\overline{AC}

, so the two pieces are equal

AE = EC

Set the expressions equal

2x + 3 = 4x - 7

Solve for

x

10 = 2x, \quad x = 5

Substitute back

AE = 2(5) + 3 = 13

Answer: $AE = 13$

Example 3: opposite sides are congruent

In parallelogram $PQRS$ , $PQ = 3x - 2$ and $SR = x + 10$ . Find $PQ$ .

\overline{PQ}

and

\overline{SR}

are opposite sides, so they are congruent

3x - 2 = x + 10

Solve for

x

2x = 12, \quad x = 6

Substitute back

PQ = 3(6) - 2 = 16

Answer: $PQ = 16$

Try one yourself

Common questions

Is a rectangle a parallelogram?

Yes. Rectangles, rhombi, and squares are all parallelograms, so every property on this page applies to them too. They just add extra properties of their own, like congruent or perpendicular diagonals.

Are the diagonals of a parallelogram congruent?

Not in general. The diagonals bisect each other — each one cuts the other in half — but the two diagonals can have different lengths. They are only congruent when the parallelogram is a rectangle (or a square).

If one angle of a parallelogram is

90^\circ

, what are the others?

All $90^\circ$ . The opposite angle matches it, and each consecutive angle must be $180 - 90 = 90^\circ$ . A parallelogram with one right angle is automatically a rectangle.

How do I keep opposite and consecutive angles straight?

Opposite angles sit diagonally across the figure and are congruent. Consecutive angles share a side and are supplementary. If you can trace one side of the parallelogram between the two angles, they add to $180^\circ$ .

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Properties of Parallelograms

The four properties

Why consecutive angles add to $180^\circ$

Matching the property to the problem

Worked examples

Example 1: finding angles

Example 2: the diagonals bisect each other

Example 3: opposite sides are congruent

Try one yourself

Common questions

Keep learning

Polygon Interior & Exterior Angle Sums

Proving a Quadrilateral Is a Parallelogram

Rectangles, Rhombi & Squares

Trapezoids & Kites

Properties of Parallelograms

The four properties

Why consecutive angles add to 180∘180^\circ180∘

Matching the property to the problem

Worked examples

Example 1: finding angles

Example 2: the diagonals bisect each other

Example 3: opposite sides are congruent

Try one yourself

Common questions

Keep learning

Polygon Interior & Exterior Angle Sums

Proving a Quadrilateral Is a Parallelogram

Rectangles, Rhombi & Squares

Trapezoids & Kites

Why consecutive angles add to $180^\circ$