Here it says that rectangles do not exist in hyperbolic geometry because if a line $l$ and a point $P$ not on $l$ are given, then there are more than one lines that passes through $P$ and parallel to $l$.
I know that the rectangles do not exist due to angle-sum theorem. If I triangulate a rectangle, then both triangles would have angle sum less than $\pi$, so the sum of angles of the rectangle would be less than $2\pi$, which is a contradiction. But I cannot see why the converse of the parallel postulate is enough to imply that there are no rectangles in hyperbolic geometry.
Best Answer
The lazy way to understand why you can't have rectangles with four right angles in a curved plane is to observe that if you have a rectangle, then using the axioms about isometries you can translate congruent rectangles, which lets you tile the plane with right-angled rectangles.
Furthermore, by bisecting pairs of sides of a rectangle, you can split the rectangle into two congruent rectangles. More generally, by making the cut not symmetric you can construct rectangles of any size. Since the tiling can be made arbitrarily fine, this effectively lets you construct Cartesian coordinates & vectors for the plane, just from the existence of a rectangle and basic notions of translating congruent polygons. You can then observe that the Cartesian plane satisfies the euclidean parallel postulate.