# 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 $\ell$ 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 .) Let $A,B,C\in\ell$ such that

• $A$, $B$, and $C$ are distinct;

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

• $A$, $B$, 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 (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 $\ell$ 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 $\overline{AC}$.

Title AAS is not valid in spherical geometry AASIsNotValidInSphericalGeometry 2013-03-22 17:13:00 2013-03-22 17:13:00 Wkbj79 (1863) Wkbj79 (1863) 8 Wkbj79 (1863) Result msc 51M10 SAA is not valid in spherical geometry