Angle Defect

Section 8.1 Angle Defect

In neutral geometry, the Saccheri–Legendre theorem tells us that the angle sum \(S\) of any triangle must be less than or equal to \(180^\circ\text{.}\) We summarize the reasoning behind this theorem below.

Lemma 8.1.1.

The sum of any two angles in a neutral triangle must be less than \(180^\circ\text{.}\)

Proof.

Consider the triangle shown in Figure 8.1.2. The angles \(\alpha\) and \(\beta\) are supplementary, so \(\alpha+\beta=180\text{.}\) But the exterior angle theorem tells us that the exterior angle \(\alpha\) must be strictly greater than the interior angle \(\gamma\text{.}\) Thus, \(\gamma \lt \alpha = 180 - \beta\text{,}\) or equivalently \(\beta+\gamma\lt 180\text{.}\)

Figure 8.1.2. The sum of any two angles in a neutral triangle must be less than \(180^\circ\text{.}\)

Lemma 8.1.3.

Given any neutral triangle, another triangle can be constructed with the same angle sum, but with any one of the angles reduced to no more than half its original size.

Proof.

Construct the midpoint \(D\) of line segment \(BC\text{,}\) then line \(AD\text{,}\) and construct \(E\) on \(AD\) so that \(D\) is also the midpoint of segment \(AE\text{,}\) as shown in Figure 8.1.4. Then triangles \(ACD\) and \(EBD\) are congruent by SAS, so the pairs of angles labeled \(\theta\) and \(\phi\) are indeed equal, as shown. Adding up the angles, we see that the angle sum of triangle \(ABE\) is the same as that of triangle \(ABC\text{.}\) But since \(\theta+\phi=\gamma\) (the original angle at point A), either \(\theta\lt\frac12\gamma\) or \(\phi\lt\frac12\gamma\) (or both). Thus, \(\triangle ABE\) is the desired triangle.

Figure 8.1.4. Shrinking one angle of a triangle while preserving the angle sum.

The rest is easy. Suppose there is a neutral triangle with angle sum greater than \(180^\circ\text{.}\) Use Lemma 8.1.3 repeatedly to make one of the angles arbitrarily small. Then the sum of the remaining two angles must exceed \(180^\circ\text{,}\) which contradicts Lemma 8.1.1.

However, we showed in Section 4.4 that if a neutral triangle with angle sum \(180^\circ\) exists, then we are in Euclidean geometry. Thus, in hyperbolic geometry, all triangles have angle sum strictly less than \(180^\circ\text{!}\)

We therefore introduce the angle defect (or simply defect) \(D\) of a hyperbolic triangle, defined by

\begin{equation} D=180-S\tag{8.1.1} \end{equation}

and note that \(D\) is strictly positive.