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I know that in Euclidean geometry, the sum of the interior angles of a triangle is exactly $\pi$.

In hyperbolic geometry, I know that the sum of the interior angles of a triangle is $\leq \pi$, and I know that there exist triangles in hyperbolic geometry with interior angles that sum to strictly less than $\pi$.

However, what I don't know is ...

How small can the sum of the interior angles of a triangle in hyperbolic geometry get? Is there a lower bound?

Is there a hyperbolic triangle with interior angle sum equal to, say, $1/10000000$? I suspect it is something nice like the angles have to add up to be at least $\pi/2$ but I don't know if that's true.

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    $\begingroup$ I can't figure out a good way to represent it, but you can have a triangle with zero angle measure. In the Half plane representation of hyperbolic geometry, this consists of a half circle of Radius $2R$ with two half circles of radius $R$ inside it, all of them tangent to each other. See this wikipedia article for visuals. $\endgroup$ Dec 18 '19 at 3:22
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    $\begingroup$ It is possible to create a triangle on the hyperbolic plane with three angles of measure 0. In fact this is an "ideal triangle." en.wikipedia.org/wiki/Ideal_triangle $\endgroup$
    – Doug M
    Dec 18 '19 at 3:24
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    $\begingroup$ @CharlesHudgins Like this? i.imgur.com/scKQ8S6.png $\endgroup$ Dec 18 '19 at 3:25
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    $\begingroup$ Yeah, although @DougM has already linked an article that has better pictures. It's a fair point that ideal triangles don't properly live in the hyperbolic plane. So the correct statement is probably that total angle measure $\theta$ satisfies $0 < \theta \leq \pi$. $\endgroup$ Dec 18 '19 at 3:27
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    $\begingroup$ In the hyperbolic plane, one can have a triangle with angles $\alpha$, $\beta$ and $\gamma$ iff $\alpha,\beta,\gamma>0$ and $\alpha+\beta+\gamma<\pi$. $\endgroup$ Dec 18 '19 at 3:39
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(Sorry if this should be a comment and not an answer: please advise). There is no lower bound. See, for example, Existence of triangles with three arbitrarily small angles in Archimedian Neutral Geometry, (e.g. in hyperbolic geometry). From baby Hartshorne for a rough sketch of a synthetic proof.

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  • $\begingroup$ Eh, probably should be a comment $\endgroup$
    – QC_QAOA
    Dec 28 '19 at 22:54
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Comment

Equilateral triangle on Poincare disc, each vertex zero (spike) angle.

Angle defect$\pi.$

enter image description here

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