Normal Euclidean geometry is on a flat surface, think a piece of paper. Where things like angles in a triangle always add up to 180 degrees.
Non-euclidean geometry is just geometry on a curved surface. Usual examples are a sphere or a horse saddle shaped plane. For example you can make a triangle with 3 90° corners, breaking the previously mentioned triangle rule.
Because of this you do things you otherwise couldn’t do in Euclidean geometry.
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