Section 6.3 Elliptic SAS
¶SAS congruence continues to hold in elliptic geometry. Just as in neutral geometry, we can illustrate SAS congruence in elliptic geometry by construction. That is, we start with an arbitrary triangle, then construct a new triangle so that two of its sides and the included angle are congruent to the corresponding parts of the original triangle. We then verify SAS congruence by measuring the remaining sides and angles in both triangles, and showing that these measurements match.
One such construction is shown in Figure 6.3.1. This particular construction will fail if the constructed triangle requires “wraparound”, that is, if constructed points appear to land outside the Klein disk. 1
It also requires both initial sides to be longer than the radius of the circle shown in the figure, although this arrangement can always be achieved by swapping these two sides if necessary.