isosceles trapezoid

isosceles trapezoid

An

isosceles trapezoid

is a

trapezoid

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 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 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 the

quadrilateral

into two congruent

right trapezoids

. In other , 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. (

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

rectangles

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

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 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

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