Angle sums in the upper half plane model

Question

I am a little confused. I see many sources state that the sum of the angles in a hyperbolic triangle is always less than $\pi.$ Yet in the upper half plane model, hyperbolic triangles are represented by triangles with arcs of circles for sides, and such triangles routinely have angle sum more $\pi,$ see e.g. https://en.wikipedia.org/wiki/Circular_triangle. So... what gives? Are triangles in the upper half plane model not actually circular triangles? Are angles in the upper half plane model measured in a much more complicated way than I thought (e.g. not just take the tangents and measure the angle)?

Answer 1

Straight line in the upper plane model are not any circles but only half circles with midpoints on the x axes , or straight vertical rays orthogonal to the x-axis . If you draw triangles with this constriction they all have angle sum <pi