a plane
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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.
A plane is a flat surface with no thickness. Our world has three dimensions, but there are only two dimensions on a plane.
Dimension is either the length, width, or height of a shape. In a 2-D or flat object there are 2 dimensions (length and width). In a 3-D shape there are 3 dimensions (length,width, and height)
When all the dimensions and angles are identical.
No they cannot. For example, in three dimensions, the angles of a triangle need not add to 180 degrees.