  We present a new model of a non-Euclidean plane, in which angles in a
triangle sum up to $\pi$. It is a subspace of the Cartesian plane over the
field of hyperreal numbers $\mathbb{R}^*$. The model enables one to represent
the negation of equivalent versions of the parallel axiom, such as the
existence of the circumcircle of a triangle, and Wallis' or Lagendre's axioms,
as well as the difference between non-Euclidean and hyperbolic planes. The
model has unique educational advantages as expounding its crucial ideas
requires only the basics of Cartesian geometry and non-Archimedean fields.
