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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.

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    A general question: Do you care what the model is? – 2017-02-16
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    I want the nicest picture, maybe the disk model will give a nice picture. Perhaps there is some natural connection with some reflection group? – 2017-02-16
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    Any hyperbolic triangle with angles of the form $\pi/k$, $\pi/m$, $\pi/n$ (with sum $<\pi$) is indeed associated with a reflection group, namely the group generated by reflections in the three sides of the triangle. Furthermore, the tiling by $\pi/k$, $\pi/m$, $\pi/n$ that is generated by the images of the original triangle under iterated reflection is invariant under the group. This is an application of the Poincare polygon theorem. – 2017-02-16
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    I think I saw a nice picture for the case of this particular reflection group/ triangle but I dont remember where, and also not the attached explanation... – 2017-02-16
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    The corresponding reflection group is also **not** on Takeuchi's [list](https://projecteuclid.org/download/pdf_1/euclid.jmsj/1240433796) of arithmetic triangle groups, so it is unlikely that this group has any number-theoretic significance. – 2017-02-17

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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 $B$.)

enter image description here

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.