Download Short notes on this section (theorem list)

Absolute geometry
The exterior angle inequality and Saccheri-Legendre
Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see section 3.4, pp 152-161.
Exterior angle inequality
An exterior angle of a triangle has angle measure greater than that of either opposite
interior angle.
[Kay, p 156]
This simple theorem is described as “the key theorem of absolute geometry.” The proof on page 156
is straightforward. It relies on the construction of a specific triangle:
and most of the proof is justifying that the triangle can be constructed and gives a pair of congruent
triangles.
After reading the proof, there’s a lemma you need to work through, that claims that the triangle
constructed above has one other property - the sum of the measures of its angles is equal to the angle
sum of the original triangle (even though none of the individual angles are congruent!)
Saccheri-Legendre theorem
The angle sum of any triangle cannot exceed 180◦ .
[Kay, p 156]
The key point here of course is that the sum isn’t necessarily equal to 180◦ , but must be less than or
equal to 180◦ .
The theorem is simple; the proof is not. There’s some fairly technical stuff going on there, but the
idea is interesting, and can be explained somewhat intuitively. See the posted lecture.