AAS is not valid in spherical geometry
AAS (http://planetmath.org/AAS) is not valid in spherical geometry^{} (http://planetmath.org/SphericalGeometry). This fact can be determined as follows:
Let $\mathrm{\ell}$ be a line on a sphere and $P$ be one of the two points that is furthest from $\mathrm{\ell}$ on the sphere. (It may be beneficial to think of $\mathrm{\ell}$ as the equator and $P$ as the .) Let $A,B,C\in \mathrm{\ell}$ such that

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$A$, $B$, and $C$ are distinct;

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the length of $\overline{AB}$ is strictly less than the length of $\overline{AC}$;

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$A$, $B$, and $P$ are not collinear^{};

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$A$, $C$, and $P$ are not collinear;

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$B$, $C$, and $P$ are not collinear.
Connect $P$ to each of the three points $A$, $B$, and $C$ with line segments^{}. (It may be beneficial to think of these line segments as longitudes.)
Since $\mathrm{\ell}$ is also a circle having $P$ as one of its centers (http://planetmath.org/Center8) with radii $\overline{AP}$, $\overline{BP}$, and $\overline{CP}$, we have that $\overline{AP}\cong \overline{BP}\cong \overline{CP}$ and that $\mathrm{\ell}$ is perpendicular^{} to each of these line segments. Thus, the triangles $\mathrm{\u25b3}ABP$ and $\mathrm{\u25b3}ACP$ have two pairs of angles congruent and a pair of sides congruent that is not between the congruent angles (actually, two pairs of sides congruent, neither of which is in between the congruent angles). On the other hand, $\mathrm{\u25b3}ABP\cong \u0338\mathrm{\u25b3}ACP$ because the length of $\overline{AB}$ is strictly less than the length of $\overline{AC}$.
Title  AAS is not valid in spherical geometry 

Canonical name  AASIsNotValidInSphericalGeometry 
Date of creation  20130322 17:13:00 
Last modified on  20130322 17:13:00 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  8 
Author  Wkbj79 (1863) 
Entry type  Result 
Classification  msc 51M10 
Synonym  SAA is not valid in spherical geometry 