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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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    Take a straightforward (counter)example: an angle inscribed in a semicircle in the hyperbolic plane.2012-04-16

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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:

enter image description here

As you can see, the angle $\alpha$ is always less than half of the angle $\beta$.

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    A is not true either. As Day Late Don suggests, this is most easily seen in the case where $\beta$ is $180^\circ$. The inscribed angle is always less than $90^\circ$, but the angle approaches $90^\circ$ as the vertex approaches one of the endpoints of the arc.2012-04-16