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As we all know, the angles in all triangles in Euclidean geometry must add up to $180^\circ$. As some of us may know, this is not true in non-Euclidean geometries; for example, on the surface of a sphere, the angles in a triangle can be between $180^\circ$ and $540^\circ$, while on a hyperbolic paraboloid (a "saddle"), it can be anywhere between $0^\circ$ and $180^\circ$. This is just one property out of many that involve triangles in non-Euclidean geometries.

A Google search on this topic brought me too broad a range of references and sources. I understand that this has been studied on and worked on a lot. What other properties of triangles exist that are contrary to normal Euclidean geometry?

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One neat example is that when considering triangles that lie on a sphere, similar triangles are congruent. That is, it is enough to know the angles of two triangles are the same to determine they are congruent because there is an absolute reference of magnitude, the radius of the sphere you consider.

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  • $\begingroup$ This is partly what makes planar graphs on spherical triangles pretty interesting. $\endgroup$ – Cameron Williams Nov 24 '14 at 2:15
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In hyperbolic geometry there exists a "horizon", and size of similar triangles changes depending on how near it is to it, or its position.

A right angled triangle in Euclidean geometry $ c^2= a^2+b^2$. In non- Euclidean hyperbolic and elliptic geometries $ \cosh c = \cosh a \cdot \cosh b , \cos c = \cos a \cdot \cos b $ for a unit radius of sphere or pseudosphere.

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