Question

(Euclidean and Non Euclidean geometry)

1. Consider the following statements:
   Given a line l and a point P not on the line:
   1. There exists at least one line through P which is perpendicular to l.
   2. There exists at most one line through P which is perpendicular to l.
   3. There exists exactly one line through P which is perpendicular to l.

Prove each statement or give a counter-example E² (Euclidean Plane), H² , (Hyperbolic Plane)and the sphere S² (Spherical Plane)

( Consider the statements separately -- don't just give a counter-example to one.)
If you give a counter-example, explain where the previous proof went wrong.
If you give a proof, explain how (or if) you had to change the first proof.

Hint: Reflect P over l.

Answer 1