This seems to be common sense, and the geometry you learned at school will have all been based on this assumption. In non-Euclidean geometry Euclid's fifth axiom is not true. This is very confusing, it seems obvious that if you have two parallel lines, then any line which intersects one of the parallel lines, will, if extended far enough, also intersect with the other and so not be parallel with it. In hyperbolic geometry however this does not happen, a line may intersect one of two parallels and never intersect the other, no matter how far it is extended. There is no trick involved in this, the line does not go into another dimension, and it stays on the paper in 2 dimensions.
Hyperbolic geometry goes counter to common sense and our experience of the world.
The best way to think of hyperbolic geometry is in terms of extremely small differences. The difference between the two intersecting lines is tiny, if you saw them both on a piece of paper you would think there was only one line. You would need to extend them for millions and millions of miles to begin to see them separating. In these less human terms it is easier to accept that they would never intersect with the third parallel line. And there is evidence that this is precisely what does happen over interstellar distances, the mathematics used in astrophysics assumes that the universe is hyperbolic over very large distances, and that the Euclidean universe we see around us is only an impression caused by our small scale.