Hyperbolic AAA

Section 8.2 Hyperbolic AAA

What are the properties of the defect defined in Section 8.1?

Consider dividing a triangle into two pieces, as shown in Figure 8.2.1, and work out the defect for all three resulting triangles. We have

\begin{gather} S(\triangle ADC) = 180 - (\theta + \phi + \delta) ,\tag{8.2.1}\\ S(\triangle ABD) = 180 - (\psi + \beta + \epsilon) ,\tag{8.2.2}\\ S(\triangle ABC) = 180 - \bigl((\theta + \psi) + \phi + \beta\bigr) ,\tag{8.2.3} \end{gather}

and it is straightfoward to verify that

\begin{equation} S(\triangle ABC) = S(\triangle ADC) + S(\triangle ABD)\tag{8.2.4} \end{equation}

using the fact that \(\delta+\epsilon=180\text{.}\)

Figure 8.2.1. Comparing the defects of divided triangles.

Defects add! The defect of a triangle is a measure of its size!

This addition principle can be extended to quadrilaterals and other polygons, replacing \(180^\circ\) by the corresponding angle sum in Euclidean geometry, such as \(360^\circ\) for quadrilaterals. Equivalently, any such polygon can be divided into triangles, for which the defects can be added.

We can now show that AAA congruence holds in hyperbolic geometry.

Wait a minute! AAA congruence? What about similar triangles? In Euclidean geometry, knowing the angles of a triangle tells you nothing about its size; you can increase all the sides proportionally, while keeping the angles unchanged. Not so in hyperbolic geometry! In fact, the addition principle still holds in Euclidean geometry, but the defects are all zero, so there is no new information. The key difference in hyperbolic geometry is that the defect is strictly positive!

Here's the outline of a proof for AAA congruence in hyperbolic geometry. Suppose you had two triangles, \(ABC\) and \(PQR\text{,}\) with the same angles but not congruent to each other. If any single pair of corresponding sides were the same length, then the triangles would be congruent by ASA (which follows from SAS). It must therefore be the case that one of the triangles has two sides each of which is longer than the corresponding side of the other triangle, as shown in Figure 8.2.2.

In this case, we can make a smaller triangle, \(P'Q'R'\text{,}\) inside this larger one (\(\triangle PQR\)) where those two sides are the same length as the corresponding sides of the first triangle (\(\triangle ABC\)). This smaller triangle (\(\triangle P'Q'R'\)) is now congruent to triangle \(ABC\) by SAS, and therefore has the same defect. But now we have two triangles, one inside the other, with the same defect – which means that the extra piece (the shaded region in Figure 8.2.2) must have zero defect, since defects add. But that's not possible; all defects are strictly positive in hyperbolic geometry.

The only way around this contradiction is for the original two triangles to be congruent, that is, for AAA congruence to hold in hyperbolic geometry.

Figure 8.2.2. The construction used to verify AAA congruence in hyperbolic geometry.

These results are illustrated in Figure 8.2.3, which allows you to explore the relationship between angle sums and area. Can you construct a hyperbolic triangle in the Poincaré disk, all of whose angles are between \(0^\circ\) and \(5^\circ\text{?}\) Do you expect this triangle to be small or large?

Figure 8.2.3. The relationship between area and angles for triangles in the Poincaré disk.