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

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

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

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

$B$, $C$, 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 $\ell$ is also a circle having $P$ as one of its centers with radii $\overline{AP}$, $\overline{BP}$, and $\overline{CP}$, we have that $\overline{AP}\cong\overline{BP}\cong\overline{CP}$ and that $\ell$ is perpendicular to each of these line segments. Thus, the triangles $\triangle ABP$ and $\triangle ACP$ 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, $\triangle ABP\not\cong\triangle ACP$ because the length of $\overline{AB}$ is strictly less than the length of $\overline{AC}$.
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