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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.

Levent
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    What is your (detailed) definition of rectangle. It is not enough to just say "a quadrilateral where, in each pair of opposite sides, the two are parallel". – Eric Towers Sep 13 '20 at 16:10
  • A quadrilateral in which every interior angle is a right angle. – Levent Sep 13 '20 at 16:12
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    Then you have already answered your own question. – Eric Towers Sep 13 '20 at 16:14
  • @EricTowers how? – Levent Sep 13 '20 at 16:14
  • Four angles, each of $\pi/2$ forces the sum of angles in your definition of a rectangle to be $2\pi$. You have shown that the sum of angles in any (convex) quadrilateral cannot be $2\pi$. – Eric Towers Sep 13 '20 at 16:15
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    Have you read the question? I already mentioned that the result follows when I use the angle-sum theorem. I want a solution using only the fact that there are more than one parallel lines that passes through a given point and a line. – Levent Sep 13 '20 at 16:17
  • @EricTowers If you follow the link you can see that the development there takes "there are no rectangles" as a lemma for which no proof is given, and derives the angle sum theorem from this. The proof is omitted with a comment that it is "technical". – Mark Bennet Sep 20 '20 at 05:47

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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.

saolof
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Comment only. Poincaré plane model hand sketch in plane but in two points (through each point pass an orthogonal pair of hyperbolic geodesics), the context is between two pairs of (hyper) parallel lines. (from a bipolar coordinate grid)

enter image description here

Narasimham
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