Trapezoid · Principia

A quadrilateral with exactly one pair of parallel sides; the parallel sides are the bases, the other two are the legs, and when the legs are equal it is an isosceles trapezoid.

Dependencies: Quadrilateral, Parallel. A derived definition: require exactly one pair of opposite sides of a quadrilateral to be parallel.

Definition

Let $ABCD$ be a quadrilateral.

Trapezoid ABCD has exactly one parallel pair, AB and DC.

If exactly one pair of opposite sides is parallel, for example $AB\parallel DC$ while $AD$ and $BC$ are not parallel, then the quadrilateral is a trapezoid. The parallel sides are its bases, and the nonparallel sides are its legs.

A trapezoid lies between a general quadrilateral and a parallelogram: it gains the order of one parallel pair but not a second.

Remarks

“Exactly one pair” is essential. If both pairs are parallel, the figure is a parallelogram, not a trapezoid under this convention.
Isosceles and right trapezoids. Equal leg lengths give an isosceles trapezoid, with equal base angles and equal diagonals (Isoceles trapezoid: base angles equal, Isoceles trapezoid: diagonals equal). A leg perpendicular to the bases gives a right trapezoid.
Midsegment. The segment joining the midpoints of the legs is parallel to both bases and has length half the sum of their lengths (Trapezoid midsegment formula).