The 18 listed below all use the hyperbolic axiom in their proof. There are many forms of non-Euclidean geometry, hyperbolic is the oldest and simplest. The fifth axiom it uses is the hyperbolic axiom, which is just the opposite of Euclid's fifth postulate.
Hyperbolic geometry goes against common sense about what straight and parallel lines are.
This can be confusing, but to understand it you have to ignore this confusion. It is possible to draw one and only one straight line from any point to any point. From each end of a finite straight line it is possible to produce it continuously in a straight line by an amount greater than any assigned length. It is possible to describe one and only one circle with any centre and radius. All right angles are equal to one another.