Section 5.4 Hyperbolic SAS
¶SAS congruence is one of the neutral postulates, so it must hold in both Euclidean and hyperbolic geometry. Just as in Euclidean geometry, we can illustrate SAS congruence in hyperbolic 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 5.4.1.
Activity 5.4.1. Verifying SAS congruence.
Can you determine how the second triangle in Figure 5.4.1 was constructed?
You may wish to compare Figure 5.4.1 with the Euclidean construction shown in Figure 2.5.1.