Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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: – 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
up vote 1 down vote accepted

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 $\beta$.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.