Thanks for the reply, I just gone on the Youtube to take a crash course into spherical trigonometry. The lectures review the basic of Euclidean geometry as an introduction to spherical geometry. Everything about Euclidean geometry sounds like just simple plane geometry I learned long time ago, but I never learn the name Euclidean geometry. I really don't know the detail, that's the reason I asked.
to me plane geometry means two dimensional geometry, either euclidean or hyperbolic, while euclidean geometry means essentially the geometry of R^n, i.e. a geometry of any finite dimension in which triangles have angle sum 180 degrees.
to people who have not studied hyperbolic plane geometry, the term plane geometry probably means the more familiar euclidean plane geometry. i do not consider the hyperbolic plane to be flat however.
I believe geometries (2D or 3D) are referred to as Euclidean if the parallel postulate holds. In Euclidean geometry, a 2D triangle has a total internal angle of 180 degrees or pi radians. In non-Euclidean geometries, there is no parallel postulate, and the total internal angle of a 2D triangle is not equal to 180 degrees.