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Two dimensional geometry is called "plane geometry" meaning that it occurs on a single surface or plane. The objects used in plane geometry are called plane figures.
There are infinitely many plane figures, not just five! A circle, ellipse, A triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, and on and on. And then there are mixed figures such as a semicircle, a segment of an ellipse. Not forgetting plane figures that have "random" boundaries.
Presumably it is simply geometry. However, plane geometry is geometry which is limited to objects with up to two dimensions, that is, objects that lie in a plane - or on a flat surface.
Geometry.
Point ; Line ; Plane - Remember the Point-Line-Plane Postulate