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I want to calculate the sum of the angles of the triangle formed in the hyperbolic plane from the points $(-1,1), (0,1)$, and $(1,1)$. This forms an angle at the origin which has an infinite slope for the hyperbolic lines going through $(1,-1)$ and $(1,1)$. What is the angle between these two points?

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You're using the upper-half-plane model, right? None of the vertices are ideal. And how does an angle have a slope? – Neal Jan 25 at 1:22
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There is no ideal angle here, you can answer your question by drawing the picture in the upper half plane as the union of three circular arcs whose intersection angles you can very easily find. – Lubin Jan 25 at 2:18

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