The parallelogram is the quadrilateral with both pairs of opposite sides parallel. Its richness lies in the fact that this single definition can be replaced by many equivalent conditions.
Definition — Parallelogram
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
Property — Necessary and sufficient conditions
A quadrilateral is a parallelogram if and only if one of the following holds:
- The opposite sides are congruent in pairs.
- One pair of opposite sides is parallel and congruent.
- The diagonals bisect each other (the midpoint is common).
- The opposite angles are congruent.
- The angles adjacent to the same side are supplementary.
- Each diagonal divides the parallelogram into two congruent triangles.
Each of these conditions, taken on its own, is enough to guarantee that the quadrilateral is a parallelogram: they are all equivalent to the definition.
Links
Topics: Euclidean geometry
Concepts: Parallelogram · Quadrilateral · Line · Parallel lines
Skills: Synthetic geometry