The diagonals divide each other into segments with lengths that are pairwise equal in terms of the picture below, AE = DE, BE = CE (and AE ≠ CE if one wishes to exclude rectangles).

Opposite angles are supplementary, which in turn implies that isosceles trapezoids are cyclic quadrilaterals.

The segment that joins the midpoints of the parallel sides is perpendicular to them.

If a quadrilateral is known to be a trapezoid, it is not sufficient just to check that the legs have the same length in order to know that it is an isosceles trapezoid, since a rhombus is a special case of a trapezoid with legs of equal length, but is not an isosceles trapezoid as it lacks a line of symmetry through the midpoints of opposite sides.Īny one of the following properties distinguishes an isosceles trapezoid from other trapezoids: However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms, crossed quadrilaterals in which opposite sides have equal length.Įvery antiparallelogram has an isosceles trapezoid as its convex hull, and may be formed from the diagonals and non-parallel sides (or either pair of opposite sides in the case of a rectangle) of an isosceles trapezoid. They can also be seen dissected from regular polygons of 5 sides or more as a truncation of 4 sequential vertices.Īny non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. Īnother special case is a 3-equal side trapezoid, sometimes known as a trilateral trapezoid or a trisosceles trapezoid. Rectangles and squares are usually considered to be special cases of isosceles trapezoids though some sources would exclude them. Special cases Special cases of isosceles trapezoids The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the supplementary angle of a base angle at the other base). In any isosceles trapezoid, two opposite sides (the bases) are parallel, and the two other sides (the legs) are of equal length (properties shared with the parallelogram), and the diagonals have equal length. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of equal measure, or as a trapezoid whose diagonals have equal length.

In Euclidean geometry, an isosceles trapezoid ( isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. Isosceles trapezoid with axis of symmetry

Volume of isosceles trapezoid prism