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5.3: Theorems of Hyperbolic Geometry

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    Note when working on these problems the model may be useful in figuring out why something works. However the proofs should be directly based on the axioms. It is legitimate to prove a statement using the model description if the model has been proven to be equivalent to the axioms. For this course proofs using the model will be worth fewer points than proofs directly from the axioms.

    Lemma: Hyperbolic Parallel Corollary

    Given a line and a point not on that line there exists infinitely many lines through that point parallel to the given line.

    Lemma: Sensed Parallel

    Given a line and a point not on that line there exists a first line through that point parallel to the given line.

    Definition: Sensed Parallel

    A line is a sensed parallel if and only if it is the first line parallel to a given line on that side through a given point.

    Definition: Angle of Parallelism

    The smaller angle formed by a sensed parallel and a transversal through the given point is the angle of parallelism if and only if the transversal is perpendicular to the given line.

    Corollary: Term

    The angle of parallelism is the same on the left and right.

    Theorem

    The angle of parallelism is less than π/2.

    Lemma

    Consider the following illustrated in \(\PageIndex{1}\). ℓ is the right sensed parallel to n at P. Let S be on ℓ to the left of P. Suppose line m through S is the sensed parallel to n at S. Show that if T is on m to the left of S then must be below TP to the right of T. Further if U on ℓ such that U-S-P and A=TP∩n, then sensed parallel must be above SA to the right of A and above UA.

    5.3.1.jpg
    Figure \(\PageIndex{1}\): Sensed Parallels Left

    Lemma: Term

    Consider the following illustrated in \(\PageIndex{2}\). ℓ is the right sensed parallel to n at P. Let S be on ℓ to the right of P. Suppose line m through S is the sensed parallel to n at S. Show that if U and T are on such that U-S-T and A=TP∩n then m is below UA to the right of U and above PA.

    5.3.2.png
    Figure \(\PageIndex{2}\): Sensed Parallels Right

    Lemma

    If ℓ is the right sensed parallel to m at P, then ℓ is the right sensed parallel to m at any point to the left of P on ℓ.

    Theorem: Sensed Everywhere

    If ℓ is the sensed parallel to m at a point P, then ℓ is the sensed parallel to m at any point Q also on ℓ.

    Definition: Omega Triangle

    A pair of parallel lines and a transversal is a omega triangle if and only if the parallels are sensed parallels.

    Draw an omega triangle using the Poincaré disc model.

    Lemma: Crossbar for Omega Triangles

    If a line k contains a vertex and an interior point of an omega triangle, it intersects the side opposite the vertex.

    Does a version of Pasch's Axiom work for omega triangles? Let ℓ and m be sensed parallels. Let AB be a transversal with A∈ℓ and B∈m Let M be the midpoint of AB and D be the foot of the perpendicular from M to ℓ. Also choose F∈m on the opposite side of AB from D such that BF≌AD Let C∈ℓ be such that C-A-D. Prove that ∠CAB≢∠ABω.

    Lemma: Omega Triangle Exterior Angle

    An exterior angle of an omega triangle is greater than the opposite interior angle.

    Lemma: Angle-Side Congruency

    Two omega triangles are congruent if the length of the finite sides and the measure of one pair of corresponding angles are congruent.

    Lemma: Angle-Angle Congruency

    Two omega triangles are congruent if corresponding pairs of angles are congruent.


    This page titled 5.3: Theorems of Hyperbolic Geometry is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Mark A. Fitch via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.