showing a non-Euclidean property in Geometry

Question

To show that in a Hyperbolic space: given a line and a point exterior this line , there are ifinity many lines that pass through this point but do not intersect the given/initial line. Where the Hyperbolic space is defined as: $H= {z : Im(z) > 0}$, it includes all the lines that are parallel to imaginary axis( or vertical to the real axis) and all the half balls that there centers are on the real axis.

Answer 1

Suppose first we are given a point $p=a+bi$ and a vertical line $e$ with points $u+xi$ with a fixed $u\in\Bbb R, \ u\ne a$ and a varying $x>0$.

Without loss of generality, assume $u>a$, and pick any $s\in (a,u)$ in the real line (which is the boundary of the upper plane model). Then the line from $p$ orthogonal to $\overline{s\, p}$ intersects the real axis at a point $t>u$, and $p$ is on the circle with center $\frac{s+t}2$ and radius $\frac{t-s}2$, by Thales' theorem.
The whole circle is strictly to the right of $e$ (its leftmost point is $s$), so doesn't intersect it, and the interval $(a,u)$ contains (continuum) infinite many points, each giving rise to a different 'line' (orthogonally intersecting semicircle).

Finally, we can either do a similar construction for the case $e$ is a semicircle, or we can use some transformation to conclude the general case.