Saccheri quadrilateral
In hyperbolic geometry, a Saccheri quadrilateral is a quadrilateral^{} such that one set of opposite sides (called the legs (http://planetmath.org/Leg)) congruent, the other set of opposite sides (called the bases (http://planetmath.org/Base9)) disjointly parallel, and, at one of the bases, both angles are right angles^{}. Since the angle sum of a triangle in hyperbolic geometry is strictly less than $\pi $ radians, the angle sum of a quadrilateral in hyperbolic geometry is strictly less than $2\pi $ radians. Thus, in any Saccheri quadrilateral, the angles that are not right angles must be acute.
The discovery of Saccheri quadrilaterals is attributed to Giovanni Saccheri.
The common perpendicular^{} to the bases of a Saccheri quadrilateral always the quadrilateral into two congruent Lambert quadrilaterals. In other , every Saccheri quadrilateral is symmetric about the common perpendicular to its bases. Thus, the two acute angles of a Saccheri quadrilateral are also congruent.
The legs of a Saccheri quadrilateral are disjointly parallel since one of the bases is a common perpendicular. Therefore, Saccheri quadrilaterals are parallelograms^{}. Note also that Saccheri quadrilaterals are right trapezoids^{} as well as isosceles trapezoids^{}.
Below are some examples of Saccheri quadrilaterals in various models. In each example, the Saccheri quadrilateral is labelled as $ABCD$, and the common perpendicular to the bases is drawn in cyan.

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The BeltramiKlein model:
In the following example, green lines indicate verification of acute angles by using the poles. (Recall that most other models of hyperbolic geometry are angle preserving. Thus, verification of angle measures is unnecessary in those models.)

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The Poincaré disc model:

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Title  Saccheri quadrilateral 

Canonical name  SaccheriQuadrilateral 
Date of creation  20130322 17:08:20 
Last modified on  20130322 17:08:20 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  16 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 51M10 
Classification  msc 5100 
Synonym  Saccheri’s quadrilateral 
Related topic  IsoscelesTrapezoid 
Related topic  RightTrapezoid 