Dear Dr. Math, I would to know if there is non-euclidean geometry that would be appropriate in difficulty for ninth graders to study. Thanks for your patience.
Euclidean geometry is the kind of geometry that assumes the Euclidean parallel postulate.
This states that given any line and any point not on that line, there is exactly one line through that point which is parallel to the given line. This is pretty amazing. It means that every pair of lines in this geometry will intersect somewhere. You can model it on the surface of a sphere, which I'll talk about later. This is geometry in which, given any line and any point not on that line, there is more than one line through that point which is parallel to the given line.