DEFINITION: Parallel lines are infinite lines in the same plane that do not intersect.

In the figure above, **Hyperbolic Line BA** and **Hyperbolic Line
BC** are both infinite lines in the same plane. They intersect
at point **B** and , therefore, they are NOT parallel Hyperbolic lines.
**Hyperbolic line DE** and **Hyperbolic Line BA** are also both
infinite lines in the same plane, and since they do not intersect, **DE**
is parallel to **BA**. Likewise, **Hyperbolic Line DE** is
also parallel to **Hyperbolic Line BC**. Now this is an odd thing
since we know that in Euclidean geometry:

- If two lines are parallel to a third line, then the two lines are parallel
to each other.

A Hyperbolic Line is not the same thing as a Euclidean Line (for example, Hyperbolic Lines curve). They do, however, share many analogous properties. The following is a list of some of these properties:

- In Euclidean geometry, there is one and only one shortest path between any two points. We call this "shortest path" the "straight" path, and this path lies along the line segment joining the two points. Exactly the same statement is true in hyperbolic geometry with Hyperbolic Points and Hyperbolic lines.
- In Euclidean geometry, two points determine a unique line. In-other-words, given any two points, there exists a line that passes through those two points. Additionally, there does not exist any other line that will pass through both of those two points. Exactly the same statement is true in Hyperbolic Geometry.
- In Euclidean geometry, light, in a vacuum, travels along a Euclidean Line. Likewise, in hyperbolic geometry, light, in a vacuum, travels along a hyperbolic line.

- In Euclidean geometry, if two lines are parallel to a third line, then the two lines are parallel to each other.
- In Euclidean geometry, if two lines are parallel then, the two lines are equi-distant.
- In Euclidean geometry, lines that do not have an end (infinite lines), also do not have a boundary (a point that they are headed toward, yet never reach).

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Next Topic - 7: Axioms and Theorems

Copyright©: Joel Castellanos, 1994-2007