Translating a sentence from one of Hilbert's papers

Question

I'm trying to translate one of Hilbert's papers. I'm 90% done, but I am stuck on one sentence, as my German is very poor and the sentence is involved. I hope someone can give me a rough\quick translation of the following lengthy sentence fragment. Thanks!

Sind nämlich irgend 2 in einer Ebene alpha gelegene und in einem Punkte C sich schneidende Geraden a und b der ursprünglichen Geometrie gegeben, so werden im Allgemeinen in jedem der 4 in alpha um C herum entstehenden ebenen Winkelräume solche gerade Linien vorhanden sein, welche keine der beiden Geraden a und b schneiden; sind jedoch insbesondere in 2 sich gegenüberliegenden ebenen Winkelräumen keine solchen geraden Linien vorhandend, so ist die fragliche Bedingung erfült;

Thanks again!

Answer 1

The sentence says something like this: If $a$ and $b$ are any two lines in the plane $\alpha$ with cutting point $C$ with respect to the original geometry, then, in general, there will exist lines in any of the 4 angle spaces around $C$, that will neither intersect $a$ nor $b$; if on the other hand there are no such lines in 2 opposite angle spaces, then the condition in question is satisfied.

Here, an "angle space" seems to be any one of the connected components of "$\alpha$ minus the union of $a$ and $b$".