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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?

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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 $180^{\circ}$.

It was conjectured for a long time that the parallel postulate (also called Euclid's fifth postulate) followed from the first four axioms of Euclidean geometry. The discovery of hyperbolic geometry showed that this wasn't the case. Hyperbolic geometry satisfies the first four postulates of Euclidean geometry but not the fifth.

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