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Question. Is it possible to construct a right triangle with a given cathetus $a$ and a given opposite angle $\alpha$ using only compass and ruler in the absolute geometry (so without the axiom of parallels)?

Observe that such a triangle always exists (by the axiom of continuity), unlike the right triangle with a given angle and given cathetus (which needs not exist in the hyperbolic geometry, but if exists, then it can be constructed by ruler and compass).

Remark. In fact, I am interested in a stronger version of this question in which it is allowed to use the compass and ruler only to construct new points as intersecting points of two lines and intersection points of circles and their diameters.

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A right triangle with a given side and opposite angle can be constructed in the hyperbolic plane with the help of a starightedge and a compass. This is done in Construction 7.2.4 of this Bachelor Thesis of Ruben de Vries.

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