Editing Spherical geometry (section)

==Properties==
Spherical geometry has the following properties:{{sfn|Meserve|1983|pp=281-282}}
* Any two great circles intersect in two diametrically opposite points, called ''antipodal points''.
* Any two points that are not antipodal points determine a unique great circle.
* There is a natural unit of angle measurement (based on a revolution), a natural unit of length (based on the circumference of a great circle) and a natural unit of area (based on the area of the sphere).
* Each great circle is associated with a pair of antipodal points, called its ''poles'' which are the common intersections of the set of great circles perpendicular to it.  This shows that a great circle is, with respect to distance measurement ''on the surface of the sphere'', a circle: the locus of points all at a specific distance from a center. 
* Each point is associated with a unique great circle, called the ''polar circle'' of the point, which is the great circle on the plane through the centre of the sphere and perpendicular to the diameter of the sphere through the given point.

As there are two arcs determined by a pair of points, which are not antipodal, on the great circle they determine, three non-collinear points do not determine a unique triangle. However, if we only consider triangles whose sides are minor arcs of great circles, we have the following properties:
* The angle sum of a triangle is greater than 180° and less than 540°.
* The area of a triangle is proportional to the excess of its angle sum over 180°.
* Two triangles with the same angle sum are equal in area.
* There is an upper bound for the area of triangles.
* The composition (product) of two reflections-across-a-great-circle may be considered as a rotation about either of the points of intersection of their axes.
* Two triangles are congruent if and only if they correspond under a finite product of such reflections. 
* Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent).