Quiz #1

In the 4-line geometry, whose axioms are:
  1. There exist exactly four lines.
  2. Any two distinct lines have exactly one point on both of them.
  3. Each point is on exactly two lines.
Prove that not all points lie on a single line.

In the proof below the justifications for the statements have not been labeled. Put the appropriate Axiom number in the spaces provided and answer the question at the end of the proof. (2 points for each correct response)

Proof:
Assume that all the points of the geometry lie on a line m.

There are (at least) two other lines, l and n, by Axiom _1____.

Each of these meets m by Axiom __2__.

The points of intersection are distinct, since otherwise there would be three lines on a point, contradicting Axiom __3__.

By Axiom _2___, l and n must have a common point and this point can not be on m, contradicting our assumption.

What type of proof is used here? (A direct proof, an indirect proof, a proof by mathematical induction, a counting proof, or a contrapositive proof) __indirect proof (or contradiction proof)___________________