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I read from different sources that the usual criteria of congruence of triangles work for spherical triangles. However it seems to me that there is a counterexample. Consider two poles on the sphere A and B. Connect them with two different spherical segments. Call C the middle point of one of these segments (the intersection with the equator). The triangle ABC has side lenghts of $\pi R$, $\frac \pi 2 R$, $\frac \pi 2 R$ which are independent on the angle $\alpha$ between the lines AC and AB. So if we thake two different values for the angle $\alpha$ the third congruence criteria is always satisfied (we have triagles with the same sides) and so the triangles should be congruent, but this is impossible since they have at least one angle that is different. Notice that also the SAS congruence criterion is failing here: these triangles have a right angle in common with equal adjacent sides. What am I missing?enter image description here

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    $\begingroup$ $ABC$ is a degenerate triangle - all vertices lie on the same line (i.e. great circle), with antipodal vertices $A$ and $B$. If you look at the proof of the theorem that says that the sides determine the congruence class you should find out where it fails in this case. $\endgroup$ – Ethan Bolker Feb 12 at 21:51
  • $\begingroup$ Also the SAS congruence criterion is failing here which is not a theorem. $\endgroup$ – Marco Disce Feb 13 at 5:47
  • $\begingroup$ Seems you are suggesting that the congruence criteria work only for nondegenerate triangles, this would be a restriction that applies to spherical geometry and not euclidean geometry. However I didn't find in my sources any restriction to the statement that the criteria are still valid for spherical triangles. $\endgroup$ – Marco Disce Feb 13 at 5:50

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