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Section 8.5 Angle Excess

Although the proof is somewhat different, all of these results carry over to elliptic geometry, but with a twist.

In elliptic geometry, the angle sum of every triangle turns out to be strictly greater than \(180^\circ\text{.}\) So instead of using defects, we define the angle excess (or simply excess) \(E\) of an elliptic triangle by

\begin{equation} E = S - 180 .\tag{8.5.1} \end{equation}

Everything else goes through as before: The excess adds, AAA congruence holds, and excess is a good measure of area (up to an overall scale).

These results are illustrated in Figure 8.5.1, which allows you to explore the relationship between angle sums and area. Can you construct an elliptic triangle in the Klein disk, all of whose angles are between \(175^\circ\) and \(180^\circ\text{?}\) Do you expect this triangle to be small or large?

Figure 8.5.1. The relationship between area and angles for triangles in the Klein disk.
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