Congruent Triangles On A Sphere
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). 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.
On the other hand, a like sequence of side, side, side (SSS) does not ensure congruence: For example, with side lengths of, and one has a continuous family of non-congruent triangles.
Read more about this topic: Spherical Trigonometry
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