Euclidian - Hyperbolic - Riemannian

"Experience proves that anyone who has studied geometry is infinitely quicker to grasp difficult subjects than one who has not." -  Plato (375 BC)

• Euclidean triangle where the sum of three angles is pi. Greek mathematician Euclid started with a system of knowledge where started with a set of postulates which are universally true and can be proved. For example "two points form a line". He started with this type of postulates then proved something else in his Euclidean geometry. It's unlike his predecessors who used axioms such as 'sky is blue'. Depending on the time of the day 'sky may be red'. So axioms are subjective and can't be proven. His masterpiece is called Elements of Geometry of simple Elements (like Issac Newton's Principia). Euclid ideas are so strong that it lasted for more than two thousand years. Until a genius named Carl Gauss suspected there is something wrong in the assumption of "Universal Truth". It formed a new type of non-euclidean geometry. The logic in the Euclidean geometry is correct but common sense says that there might be another kind of geometry where the Euclidean assumptions are not valid.
 

• Hyperbolic triangle has a total angle which is less than pi. The amount by which it is less than pi is called the defect. It has its implication in Einstein's general theory of relativity. It's also called Labochevskian geometry. Escher painted famous artwork that involves hyperbolic geometry in it.

• Riemannian triangle has a total angle which is greater than pi. This triangle can be seen on a spherical surface eg, earth. Straight-line on this surface is curved. Application in Geodesic computation. Einstein uses Riemannian geometry in his general theory of relativity to show the space-time curves.

Reference: Gaussian_curvature
Reference: Hyperboloid