Question

prove x axis and y axis are perpendicular to each other

Answer 1

Because they are the easiest to deal with , you can choose any other pair of lines which are not perpendicular and also not parallel to describe the whole R^2 plane but in that case you have to deal with parallelogram instead of rectangle and a triangle instead of right angle as we have in XY co-ordinate system.

Answer 2

They are considered as perpendicular, since the xy plane must represent a two dimensional surface. To represent anything on 2D plane (Cartesian coordinate), you need two values, distance of the "thing" from each edge (each edge is a 1-D scale, both edges participate together to form what we call 2D), of the reference scale.
Similarly in a n-dimensional space, each axis is perpendicular to each other, though it is difficult to visualize dimensions more than 3D.