Properties of Parallelogram — Sides, Angles, Diagonals

TL;DR
A parallelogram is a quadrilateral with both pairs of opposite sides parallel, and that single rule forces every other property: opposite sides are equal, opposite angles are equal, consecutive angles add to 180°, and the diagonals bisect each other. This guide proves each property, lists the formulas, and works through six examples.

What Are The Properties of a Parallelogram?

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. From that one definition, four properties follow for the sides, angles, and diagonals, and they are not separate facts to memorise. They all chain back to the parallel-sides rule through a single idea: a diagonal splits the shape into two congruent triangles.

The core properties are:

Opposite sides are equal in length, and both pairs are parallel.
Opposite angles are equal (∠A=∠C, ∠B=∠D).
Consecutive (adjacent) angles are supplementary: each neighbouring pair sums to 180°.
The diagonals bisect each other: they cut each other into two equal halves at their crossing point.

Side Properties: Why Opposite Sides Are Equal

The definition only promises that opposite sides are parallel. That they are also equal is something you prove. Draw diagonal AC. It cuts parallelogram ABCD into triangles ABC and CDA.

In those two triangles:

∠BAC=∠DCA (alternate interior angles, since AB is parallel to DC).
∠BCA=∠DAC (alternate interior angles, since AD is parallel to BC).
Side AC is shared by both triangles.

By the Angle-Side-Angle rule, triangle ABC is congruent to triangle CDA. Matching sides of congruent triangles are equal, so AB=DC and BC=AD. Opposite sides are equal, proved rather than assumed.

Theorem 1 (and its converse): In a quadrilateral, if one pair of opposite sides is both equal and parallel, the quadrilateral is a parallelogram. This converse is the test engineers and students use most: you do not need to check all four sides, just one matched pair.

Angle Properties: Opposite Equal, Consecutive Supplementary

The same congruent triangles deliver the angle facts. Because triangle ABC is congruent to triangle CDA, the matching angles ∠B and ∠D are equal. Repeat with the other diagonal BD and you get ∠A=∠C. So opposite angles are equal.

Consecutive angles are a different relationship. Side AD crosses the two parallel lines AB and DC, so ∠A and ∠D are co-interior angles (same-side interior angles), which always sum to 180°.

∠A+∠D=180°

The same holds for every neighbouring pair. A neat consequence: if any one angle of a parallelogram is 90°, its consecutive angle is 90° and then all four are 90°, which is precisely how a parallelogram becomes a rectangle.

Diagonal Properties: The Diagonals Bisect Each Other

The most-tested property: the two diagonals of a parallelogram cut each other exactly in half. Let the diagonals AC and BD meet at O.

Look at triangles AOB and COD:

AB=CD (opposite sides equal, just proved).
∠OAB=∠OCD (alternate interior angles, AB parallel to CD).
∠OBA=∠ODC (alternate interior angles).

By Angle-Side-Angle, triangle AOB is congruent to triangle COD. So AO=OC and BO=OD, and the diagonals bisect each other at O.

Two cautions worth stating plainly. The diagonals bisect each other, but in a general parallelogram they are not equal and they do not meet at 90°. Diagonals become equal only when the shape is a rectangle, and perpendicular only when it is a rhombus.

Formulas For A Parallelogram

Each formula below lists what its symbols mean.

Area, base b and perpendicular height h: Area=b×h
Perimeter, adjacent side lengths a and b: Perimeter=2(a+b)
Parallelogram law, relating the four sides to the two diagonals d1 and d2: d1²+d2²=2(a²+b²)

That last identity is the one students meet last and forget first, so here is where it comes from: apply the law of cosines to the two triangles a diagonal makes, add the results, and the cosine terms cancel because consecutive angles are supplementary. The sum of the squares of the diagonals equals twice the sum of the squares of the sides.

How Is A Parallelogram Different From A Rectangle, Rhombus, or Square?

Each of those three is a parallelogram with one extra constraint switched on, so each inherits every property above and adds its own.

Shape	Extra constraint	New diagonal behaviour
Rectangle	All angles 90°	Diagonals become equal
Rhombus	All sides equal	Diagonals become perpendicular
Square	Both at once	Diagonals equal and perpendicular

This inheritance is the practical payoff of the whole topic. Prove "diagonals bisect each other" once for the parallelogram, and rectangles, rhombuses, and squares get it for free.

Examples of Properties of Parallelogram

Example 1

In parallelogram ABCD, ∠A = 70°. Find the other three angles.

Opposite angles are equal, so ∠C=∠A=70°.

Consecutive angles are supplementary: ∠B=180°−70°=110°, ∠D=∠B=110°.

Final answer: ∠B = ∠D = 110°, ∠C = 70°.

Example 2

The diagonals of a parallelogram are claimed to be equal because "the two triangles look the same." A diagonal AC measures 10 cm and BD measures 14 cm. Is the claim right?

The error is mixing up two different properties. Bisecting each other (always true) is not the same as being equal (only true in a rectangle).

Final answer: The claim is wrong; diagonals bisect each other but are not equal unless the parallelogram is a rectangle.

Example 3

One pair of opposite sides of quadrilateral PQRS is both parallel and equal. Is PQRS a parallelogram?

Yes, by the converse of Theorem 1: if one pair of opposite sides is both equal and parallel, the quadrilateral is a parallelogram.

Final answer: Yes, PQRS is a parallelogram.

Example 4

A parallelogram has base 9 cm and perpendicular height 4 cm. Find its area. Area=b×h Area=9×4=36 cm². Final answer: 36 cm².

Example 5

In parallelogram ABCD, the diagonals meet at O. If AO = 5 cm and BO = 7 cm, find the full lengths of both diagonals.

AC=2×AO=10 cm, BD=2×BO=14 cm. Final answer: AC = 10 cm, BD = 14 cm.

Example 6

A parallelogram has sides 6 cm and 8 cm, and one diagonal of 10 cm. Use the parallelogram law to find the other diagonal.

Final answer: The other diagonal is 10 cm. (Equal diagonals here signal this particular parallelogram is a rectangle.)

Why The Properties of A Parallelogram Matter

The defining property, that opposite sides stay parallel, is the one that gets used in the physical world. A pantograph, the linkage that copies and scales drawings, and the parallel-rule drafting tool both rely on a four-bar parallelogram so that a moving arm stays parallel to a fixed one no matter how the joints swing.

The proofs matter for a quieter reason. Geometry is where most students first meet the idea that a claim has to be earned from prior facts, not just observed in a drawing. That distinction, between what looks true and what is shown true, is the actual skill the parallelogram is teaching, and it carries straight into every theorem after.

Tripping Points To Avoid

Mistake 1: Assuming the diagonals are equal

The correct way: Diagonals bisect each other in every parallelogram, but are equal only in a rectangle.

Mistake 2: Using the slant side as the height

The correct way: The height in Area=b×h is the perpendicular distance between the parallel sides.

Mistake 3: Confusing consecutive angles with opposite angles

The correct way: Opposite angles are equal; consecutive angles sum to 180°.

Conclusion

The properties of a parallelogram all flow from one rule: both pairs of opposite sides are parallel.
Opposite sides are equal, opposite angles are equal, and consecutive angles are supplementary.
The diagonals bisect each other, but are not equal (rectangle only) and not perpendicular (rhombus only).
The converse test (one pair of sides equal and parallel) is the quickest way to confirm a parallelogram.