Editing Congruence (geometry) (section)

==Congruent triangles on a sphere==
{{Main|Solving triangles#Solving spherical triangles|Spherical trigonometry#Solution of triangles}}

As with plane triangles, on a sphere two triangles sharing the same sequence of angle-side-angle (ASA) are necessarily congruent (that is, they have three identical sides and three identical angles).<ref name=Bolin>{{cite web |first=Michael |last=Bolin |title=Exploration of Spherical Geometry |date=September 9, 2003 |pages=6–7 |url=http://math.iit.edu/~mccomic/420/notes/Bolin_spherical.pdf#page=6 |archive-url=https://ghostarchive.org/archive/20221009/http://math.iit.edu/~mccomic/420/notes/Bolin_spherical.pdf#page=6 |archive-date=2022-10-09 |url-status=live }}</ref> This can be seen as follows: One can situate one of the vertices with a given angle at the south pole and run the side with given length up the prime meridian. Knowing both angles at either end of the segment of fixed length ensures that the other two sides emanate with a uniquely determined trajectory, and thus will meet each other at a uniquely determined point; thus ASA is valid.

The congruence theorems side-angle-side (SAS) and side-side-side (SSS) also hold on a sphere; in addition, if two spherical triangles have an identical angle-angle-angle (AAA) sequence, they are congruent (unlike for plane triangles).<ref name=Bolin/>

The plane-triangle congruence theorem angle-angle-side (AAS) does not hold for spherical triangles.<ref>{{cite web |last=Hollyer |first=L. |url=http://www.uh.edu/~hollyer/Module6/m6ppt/sld089.htm |title=Slide 89 of 112 }}</ref> As in plane geometry, side-side-angle (SSA) does not imply congruence.