## You are here

Homeisosceles trapezoid

## Primary tabs

# isosceles trapezoid

An *isosceles trapezoid* is a trapezoid whose legs are congruent and that has two congruent angles such that their common side is a base of the trapezoid. Thus, in an isosceles trapezoid, *any* two angles whose common side is a base of the trapezoid are congruent.

In Euclidean geometry, 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 essential for the definition of an isosceles trapezoid, as the other requirements do not imply that the legs are congruent.

The common perpendicular bisector to the bases of an isosceles trapezoid always divides the quadrilateral into two congruent right trapezoids. In other words, every isosceles trapezoid is symmetric about the common perpendicular 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. British English), this figure is referred to as an *isosceles trapezium*. Because of the modifier “isosceles”, no confusion should arise with this usage.

All rectangles are isosceles trapezoids (unless the restricted definition of trapezoid is used, see the entry on trapezoid for more details). Note that, in Euclidean geometry, if a parallelogram is an isosceles trapezoid, then it must be a rectangle.

In Euclidean geometry, in a circle, the endpoints of two parallel 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 right isosceles trapezoid 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 current usage.

## Mathematics Subject Classification

51-00*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith