Theorem 5.4.1.
If two lines are sensed parallels to a third line, they are also sensed parallels to each other.
Section 5.4 Parallels in Hyperbolic
Theorem 5.4.2.
If two lines are sensed parallels to a third line, the line farthest away has the smallest angle of parallelism.
Proof.
The theorem follows from the hyperbolic exterior angle theorem. Let \(m\) and \(n\) be distinct sensed parallels to \(l\text{,}\) with \(m\) strictly between \(n\) and \(l\text{.}\) Let \(t\) be a transversal of these three lines perpendicular to \(l\) at point \(A\text{.}\) Let \(B = m\cap t\) and \(C = n\cap t\text{.}\) Then \(AB\Omega\) and \(BC\Omega\) are omega triangles, and \(\angle OmegaBA\) is the angle of parallelism of \(n\) with respect to \(l\text{.}\) By the vertical angle theorem, this angle is also an exterior angle of \(BC\Omega\text{,}\) so we must have \(m\angle AB\Omega > \angle BC\Omega\text{.}\) But \(\angle BC\Omega\) is the angle of parallelism of \(n\) with respect to \(l\text{.}\) So the angle of parallelism of the further sensed parallel is smaller than that of the closer sensed parallel.
Definition 5.4.3. Saccheri Quadrilateral.
A quadrilateral is a Saccheri quadrilateral if and only if it has two consecutive right angles adjacent to two congruent sides. The side orthogonal to two sides is the base. The opposite side is the summit.
Theorem 5.4.4.
The non-right angles in a Saccheri quadrilateral are congruent.
Theorem 5.4.5.
The line segment joining the midpoint of the base to the midpoint of the summit is orthogonal to both.
Lemma 5.4.6.
Let \(ABCD\) be a Saccheri quadrilateral with right angles at \(A\) and \(B.\) Prove that \(\angle AD\Omega \cong \angle BC\Omega.\)
Theorem 5.4.7.
The non-right angles in a Saccheri quadrilateral are acute.
Theorem 5.4.8.
Parallel lines are not everywhere equidistant.
Theorem 5.4.9.
A transversal perpendicular to two parallel lines is unique.
