# Finding an angle of a triangle in the upper half plane model given three points

I've been given three points in the upper half plane $(i, 3i, 1 + 2i$), and one of the homework questions that asked is to find the angles of the given triangle. A previous problem asks to find the length of the sides of the triangle, and then I'm also asked to find the measure of the angles of the triangle. Would they relate? if they do not, how would I find the angles?

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Hi, leinaudGnus, and welcome to Math.SE! It appears the points you ask about are in the complex plane, and they form a triangle (as they do not lie in a common line). I'd help to improve your wording, but I notice that the tag for hyperbolic-geometry was chosen by you. Is there an issue about hyperbolic geometry here, or is it simply plane geometry? – hardmath Apr 29 '13 at 0:13
Is the the Euclidean complex plane, or hyperbolic plane? I see they hyperbolic geometry tag, but it doesn't say hyperbolic in the question. – Ross Millikan Apr 29 '13 at 20:43
Given that he talks about the triangle living in the upper half plane, and specifically calls it the upper half plane model in the title, it's safe to assume he's talking about hyperbolic geometry here. – Dan Rust Apr 29 '13 at 22:12

Recall that the geodesics in the upperhalf plane model of hyperbolic geometry, $H$, are semicircles centered on the real line and vertical straight lines with constant real value. Try to find the circular/vertical arcs/line segments which connect your three points.
Another point you might like to recall is that the upperhalf plane model of hyperbolic geometry actually preserves angles from the Euclidean geometry of the complex plane $\mathbb{C}$. This means that if you can find the angles of the triangle at the point of intersection in the euclidean case, with the same curved edges *, then the angles will be the same in the hyperbolic case, and so you're done. Note that in hyperbolic geometry, there exist triangles with zero angles (although any such vertices with zero angle lie on the boundary of $H$, so don't apply to this problem).
When i say in the Euclidean case, I mean the same curves in $H$ which give the hyperbolic triangle (so in your case 2 arcs of a semi-circle and a vertical line segment) but you measure the angles as you normally would in the Euclidean case, by comparing the tangent lines at the intersection of two curves. – Dan Rust Apr 29 '13 at 21:12