Show that for every angle in the hyperbolic plane $\mathbb{H}^2$ there is an associated line, which is completly in the interior of the angle.

Question

As I'm learning for my upcoming geometry exam, I would appreciate if someone could assist me with this exam question. I'm not really sure where to start :/. Thanks in advance!

Edit: Here is a geometric representation of what is supposed to be shown.

In the hyperbolic plane there is a line g, which is in the interior of the angle, which is created by the noncollinear points A, B and C. The statement is supposed to be proven using the half-plane model.

Answer 1

You may assume that you are dealing with the Poincare model ( unit disk). Assume that $A$ is in fact in the center of the euclidian unit circle. Then the hyperbolic lines $AB$, $AC$ are straight euclidian lines. Let them intersect the unit circle at $B'$, $C'$. Now draw the hyperbolic line passing through the absolute points $B'$, $C'$ ( it will be an arc of circle orthogonal to the unit circle, and passing through $B'$, $C'$. Here is a line that does not intersect the angle. Now if instead of $B'$, $C'$ you take other points $B"$, $C"$ on the unit circle inside the arc $B'$, $C'$, and take the hyperbolic line through them, again you get a hyperbolic line strictly inside your angle.