# Sum of Interior Angles of a Triangle

I was just wondering,is there a way to prove that the sum of the interior angles of a triangle add up to $180^{\circ}$ without using the parallel postulate?

• – The Chaz 2.0 Sep 23 '13 at 4:21

No. The fact that the sum of the interior angles of a triangle add up to $180^{\circ}$ is equivalent to the parallel postulate.
Without the parallel postulate, one obtains non-Euclidean geometries, namely hyperbolic and elliptic geometries. In these alternative geometries, the interior angle sum of a triangle is not $180^{\circ}$. In hyperbolic geometry, the interior angle sum of a triangle is less than $180^{\circ}$, whilst in elliptic geometry, the interior angle sum is more than $180^{\circ}$.