Hyperbolic triangle with angles $\pi/2$, $\pi/6$ and $\pi/9$.

Question

I need (inspiring) drawings of a hyperbolic triangles with angles $\pi/2$, $\pi/6$ and $\pi/9$ and some information if such a triangle emerges naturally in some interesting mathematical contexts.

Answer 1

I would suggest to use the Poincare disk model in this case because it depicts angles like they were Euclidean angles.

The following figure depicts a triangle with angles $90^{\circ}$, $60^{\circ}$, and $0^{\circ}$. (not $30^{\circ}$ because the figure would have become a little overcomplicated. However you could halve the angle at $B$.)

If you decrease the length of $AB$ then the third angle will increase from $0$ to $30^{\circ}$. (Note that if the triangle is getting smaller and smaller then the laws of Euclidean geometry will be better and better approximations.) In between (between $0$ and $30$) somewhere the third angle will be of $20^{\circ}$

I downloaded the NonEuclid software from here and did my construction with the NonEuclid software. You can do the same construction (plus halving the angle at $B$) and watch how the third angle changes while changing the length $AB$.

Note that in hyperbolic geometry there is no similarity. That is all the triangles with the same triplets of angles are congruent.