Straightedge and compass constructions and absolute geometry

Straightedge and compass constructions only use Euclid's first three axioms (that is "a line connects two points," "a line can be extended forever," and "a circle is defined by its center and radius").

As the parallel postulate is not used here, does that tell us that anything we can construct holds in absolute geometry as well?

Some simple examples:

1. We can bisect any angle
2. We can construct equilateral triangles
3. We can construct a perpendicular bisector to a line segment

It makes sense that all three of these things would work in absolute geometry.

That said, I can also easily build a square (equal sides and right angles) using a straightedge and compass as well, but squares don't exist hyperbolic geometry (because I can't have a quadrilateral with four right angles). What am I missing here?

• In the construction of the square you assume that the fourth line, perpendicular to the third one, will also be perpendicular to the second line. But that needn't be the case in absolute geometry. Jul 12 at 15:28
• Yes! That makes sense. Thank you! Jul 12 at 16:15