isosceles trapezoid


An isosceles trapezoidMathworldPlanetmath is a trapezoidMathworldPlanetmath whose legs (http://planetmath.org/Leg) are congruent and that has two congruent angles such that their common side (http://planetmath.org/Side3) is a base of the trapezoid. Thus, in an isosceles trapezoid, any two angles whose common is a base of the trapezoid are congruent.

In Euclidean geometryMathworldPlanetmath, the convention is to state the definition of an isosceles trapezoid without the condition that the legs are congruent, as this fact can be proven in Euclidean geometry from the other requirements. For other geometries, such as hyperbolic geometry and spherical geometry, the condition that the legs are congruent is for the definition of an isosceles trapezoid, as the other requirements do not imply that the legs are congruent.

The common perpendicular bisectorMathworldPlanetmath to the bases of an isosceles trapezoid always the quadrilateralMathworldPlanetmath into two congruent right trapezoidsMathworldPlanetmath. In other , every isosceles trapezoid is symmetric about the common perpendicularMathworldPlanetmathPlanetmathPlanetmath to its bases.

Below is a picture of an isosceles trapezoid. The common perpendicular to its bases is drawn in cyan.

In some dialects of English (e.g. (http://planetmath.org/Eg) British English), this figure is referred to as an isosceles trapezium. Because of the modifier “isosceles”, no confusion should arise with this usage.

All rectanglesMathworldPlanetmathPlanetmath are isosceles trapezoids (unless the definition of trapezoid is used, see the entry on trapezoid (http://planetmath.org/Trapezoid) for more details). Note that, in Euclidean geometry, if a parallelogramMathworldPlanetmath is an isosceles trapezoid, then it must be a rectangle.

In Euclidean geometry, in a circle, the endpointsMathworldPlanetmath of two parallelMathworldPlanetmathPlanetmath chords are the vertices of an isosceles trapezoid. Conversely, one may use four suitable points on a circle for obtaining parallel chords (and thus parallel lines).

A is a trapezoid that is simultaneously a right trapezoid and an isosceles trapezoid. In Euclidean geometry, such trapezoids are automatically rectangles. In hyperbolic geometry, such trapezoids are automatically Saccheri quadrilaterals. Thus, the phrase “right isosceles trapezoid” occurs rarely.

A 3-sides-equal trapezoid is an isosceles trapezoid having at least three congruent sides. Below is a picture of a 3-sides-equal trapezoid.

In some dialects of English (e.g. British English), this figure is referred to as a 3-sides-equal trapezium. Because of the modifier “3-sides-equal”, no confusion should arise with this usage.

A rare but convenient alternative name for a 3-sides-equal trapezoid is a trisosceles trapezoid; the corresponding name trisosceles trapezium does not seem to be in usage.

Title isosceles trapezoid
Canonical name IsoscelesTrapezoid
Date of creation 2013-03-22 17:11:59
Last modified on 2013-03-22 17:11:59
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 25
Author Wkbj79 (1863)
Entry type Definition
Classification msc 51-00
Synonym isosceles trapezium
Related topic SaccheriQuadrilateral
Defines 3-sides-equal trapezoid
Defines 3 sides equal trapezoid
Defines 3-sides-equal trapezium
Defines 3 sides equal trapezium
Defines trisosceles trapezoid
Defines trisosceles trapezium