Properties of Parallelograms

A parallelogram is defined by one thing: both pairs of opposite sides are parallel. Everything else follows.

From that single definition you get:

• opposite sides congruent
• opposite angles congruent
• consecutive angles supplementary
• diagonals bisect each other

That last one is easy to misremember. The diagonals cut *each other* in half, but they are not equal to each other and they do not bisect the angles.

Each of these has a converse that proves a quadrilateral *is* a parallelogram. Showing opposite sides congruent, or diagonals bisecting each other, or one pair of sides both parallel and congruent, is enough.

That last test — one pair both parallel and congruent — is often the fastest in coordinate geometry.

From the definition (both pairs of opposite sides parallel), the properties follow by drawing a diagonal and applying ASA: the diagonal creates two triangles congruent by alternate interior angles and the shared side, and CPCTC then gives congruent opposite sides and angles.

Consecutive angles are supplementary because they are same-side interior angles between parallel lines.

Each property's converse is a valid test for a parallelogram: both pairs of opposite sides congruent; both pairs of opposite angles congruent; diagonals bisecting each other; or one pair of opposite sides both parallel and congruent. Note that one pair parallel and the *other* pair congruent is not sufficient — that admits an isosceles trapezoid.

In coordinate proofs, the efficient route is usually the midpoint test: computing the midpoints of both diagonals and showing they coincide proves the diagonals bisect each other in two calculations, with no slopes or distances required.

A parallelogram's diagonals also divide it into four triangles of equal area, and each diagonal alone bisects its area.