Section 4.5 Hyperbolic Geometry
¶In Section 4.3, we argued that the exterior angles of a triangle must be larger than the corresponding nonadjacent interior angles. We can use this result to give a condition that ensures two lines are parallel.
Claim 4.5.1.
Suppose two lines are crossed by a third such that a pair of alternate interior angles are equal. Then the two lines are parallel.
Proof.
If the two lines intersect, we would have the situation shown in Figure 4.5.2, in which one of the angles is interior and the other exterior to the resulting triangle. But this violates the aforementioned result that the exterior angle must be larger than the interior angle.
It is now straightforward to show that parallel lines exist in neutral geometry.
Claim 4.5.3.
Given any line and any point not on the line there exists at least one line through the given point parallel to the given line.
Proof.
Construct the perpendicular line from the given point to the given line, then construct the line through the given point that is perpendicular to the line just constructed. Since the interior angles are all \(90^\circ\text{,}\) alternate interior angles are equal, so this latter line must be parallel to the original ine. See Figure 4.5.4.
So parallel lines exist in neutral geometry. But how many?
One possibility is for the Euclidean parallel postulate to hold, with exactly one line through a given point parallel to a given line. Suppose that the Euclidean parallel postulate does not hold. Then there must be more than one line through a given point parallel to a given line, which is the hyperbolic parallel postulate. It is not hard to show that in fact there are infinitely many such parallel lines, as we will see once we have introduced a model for hyperbolic geometry in the next chapter.