# 5.4: Parallels in Hyperbolic

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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.

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.