We all know that in Euclidean geometry a) the inscribed angle is always the same b) it's half of the central angle. Can we prove either of these without presuming the parallel postulate?
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The measure of an inscribed angle in the hyperbolic plane is always less than half the measure of the central angle. Here is a picture using the Poincaré disk model:
As you can see, the angle $\alpha$ is always less than half of the angle $\beta$.