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5.4: Parallels in Hyperbolic

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    89860

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    Theorem

    If two lines are sensed parallels to a third line, they are also sensed parallels to each other.

    Theorem

    If two lines are sensed parallels to a third line, the line farthest away has the smallest angle of parallelism.

    Definition: 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

    The non-right angles in a Saccheri quadrilateral are congruent.

    Theorem

    The line segment joining the midpoint of the base to the midpoint of the summit is orthogonal to both.

    Lemma

    Let ABCD be a Saccheri quadrilateral with right angles at A and B. Prove that ∠ADΩ≌∠BCΩ.

    Theorem

    The non-right angles in a Saccheri quadrilateral are acute.

    Theorem

    Parallel lines are not everywhere equidistant.

    Theorem

    A transversal perpendicular to two parallel lines is unique.

    This page titled 5.4: Parallels in Hyperbolic 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.