Conditions for a Parallelogram

Knowing what a parallelogram has is one thing; proving that some quadrilateral is one is another. Four theorems do that job. Each names a single condition which, once verified, is enough on its own — you never need to check the definition directly.

Concept Property or condition?

The two words point in opposite directions, and mixing them up is the single most common error in this topic.

Property — you already know it is a parallelogram, so you may conclude the sides are equal.

Condition — you know the sides are equal, so you may conclude it is a parallelogram.

This lesson is about the second direction. Each theorem below starts from something you can measure and ends at the shape's name.

Theorem Both pairs of opposite sides congruent

If each pair of opposite sides of a quadrilateral is congruent, the quadrilateral is a parallelogram.

If and , then is a parallelogram.

Single and double ticks distinguish the two pairs. Note that both pairs are required — one pair of equal sides on its own proves nothing.

Theorem Both pairs of opposite angles congruent

If each pair of opposite angles of a quadrilateral is congruent, the quadrilateral is a parallelogram.

If and , then is a parallelogram.

Here the congruent sets are marked by arc count rather than tick count — one arc for the first pair, two for the second.

Theorem The diagonals bisect each other

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

If and bisect each other, then is a parallelogram.

Bisecting means each diagonal cuts the other exactly in half, so all four half-segments meet at one point. The diagonals need not be equal in length, and they need not meet at a right angle.

Theorem One pair both parallel and congruent

If one pair of opposite sides is both parallel and congruent, the quadrilateral is a parallelogram.

If and , then is a parallelogram.

This is the most economical of the four: a single pair of sides settles the matter, provided that pair is both parallel and equal. Parallel alone gives only a trapezium; equal alone is not enough either.

Example Choosing the right condition

In quadrilateral , the diagonals meet at with , , and . Is a parallelogram?

, so

is the midpoint of

, so

is the midpoint of

Each diagonal therefore cuts the other in half

Yes — by the diagonal condition, is a parallelogram

The diagonals here are and , so they are certainly not equal — and that does not matter. The condition asks only that each be bisected.

Summary

A condition proves a shape is a parallelogram; a property describes one you already have.
Both pairs of opposite sides congruent parallelogram.
Both pairs of opposite angles congruent parallelogram.
Diagonals that bisect each other parallelogram.
One pair of sides both parallel and congruent parallelogram.
Any one condition is sufficient by itself — verify a single one and the proof is complete.