I would modify the wording of @Aretino's comment:
"that a [line not having common points] to a line passing through a given point exists is a theorem in absolute geometry."
A line having no common points to another line is called an ultra parallel line. With this the sentence above reads:
"That an ultra parallel line to a line passing through a given point exists is a theorem in absolute geometry."
In the Euclidean geometry the following concepts are equivalent: ultra parallel, equidistant, parallel.
In the hyperbolic geometry parallel or asimptotically parallel and equidistant and ultra parallel are different concepts.
Consider the following figure:
In absolute geometry: If we drop a perpendicular to $l$ from $P$ and then another perpendicular to this latter line then we will get a line $l'$ that will be ultraparallel to $l$. That is, a line not having common points to $l$.
Since this is an absolute theorem, it is true both in Euclidean and hyperbolic geometries.
In hyperbolic geometry: $l'$ constructed the way given above will be simply an ultra parallel line.
In Euclidean geometry the said line will be equidistant to $l$.
How to construct the asimpptotitically parallel lines in hyperbolic geometry? The following is the Bolyai construction:
Drop a perpendicular from $P$ to $l$ and construct $l'$, a perpendicular line at $P$ to $PA$. Then take an arbitrary point $B\not =A$ on $l$. Drop a perpendicular from $B$ to $l'$. Draw a circle of radius $AB$ around $P$. Construct the straights joining $P$ and the intersection points of the circle and $b$.