In 19th century we have the concept of Non Euclidean geometry that is divided into two parts spherical geometry and hyperbolic geometry. Euclidean plane is not quite the same as the Cartesian plane. In geometry a plane is a flat, two dimensional surface that extends infinitely far but has no thickness. In reality it is bit difficult to visualise or think a plane. The actual concept of plane is imaginary on the surface of the earth. In broader sense it is no longer true that the sum of a triangle is 180 degree, however to some extent it may be true for very small triangle because from the perspective of a very small triangle, the surface of a sphere is nearly flat otherwise it is pertinent to consider for a spherical triangle (A spherical triangle formed by the arcs of the three great circles each of whose poles is the vertex of a given spherical triangle) or hyperbolic triangle (A hyperbolic triangle is just three points connected by the hyperbolic line segment) the sum of the three interior angles is less than or more than 180 degree respectively.
Euclidean plane, Cartesian plane, Triangle, Sphere, and Hyperbola.
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