# Inscribed angle is always the same and twice the central angle — is this absolute?

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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Is the question "Can we prove a), b) outside of Euclidean geometry"? Also, the body and subject don't match. – The Chaz 2.0 Apr 16 '12 at 20:00
@The Chaz: en.wikipedia.org/wiki/Absolute_geometry – Rahul Apr 16 '12 at 20:01
Take a straightforward (counter)example: an angle inscribed in a semicircle in the hyperbolic plane. – Blue Apr 16 '12 at 20:36

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 $\beta$.

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OK so B is not true, what about A? Is $\beta-\alpha$ a constant for the same arc (even if not $\beta\over 2$)? – chx Apr 16 '12 at 23:10
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. – Jim Belk Apr 16 '12 at 23:32