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Why is SAS (Side-Angle-Side) Congruency holds true on a plane but not on a sphere? I am trying to understand why it is so (instead of proving/disproving the same). Please help me understand.

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"Why is this true?" is standard English grammar, and "Why does this hold true?" is also standard. "Why does this holds true?" is an error that seems to me to be frequently made by French-speaking people. –  Michael Hardy Jul 8 '12 at 18:23
    
What's your definition of a triangle? Must all the sides be length minimizing geodesics? Are you allowed to have angles of $\pi$ radians? –  Jason DeVito Jul 8 '12 at 19:38

3 Answers 3

If I'm not mistaken, if two triangles on a sphere are in the SAS relation, then they are congruent.

But if they're on two different spheres, of different sizes, then they're not.

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SAS congruence is true for spherical triangles. See Todhunter's book (I think it is freely available on internet)

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In the comments, Jason raised the question of the legitimacy of angles of $\pi$ radians. This may be what Jason had in mind:

Let $A,B$ be antipodal, let $C$ be any other point. Then there are infinitely many non-congruent triangles $ABC$, all sharing the sides $AC$ and $BC$ and the angle at $C$. Of course, that angle at $C$ is $\pi$.

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