Two distinct intersecting lines in a plane have exactly how many points in common?

A proof builds a chain of statements, each justified by something already accepted.

**Example**: Two distinct lines cannot have more than one point in common.

**Proof**: Suppose two distinct lines ℓ and m had two points A and B in common.

- By Postulate 1, only one straight line can be drawn through A and B.
- So ℓ and m must be the same line — contradicting that they are distinct.
- Hence our supposition is false. ∎

This is **proof by contradiction** (reductio ad absurdum). Assume the opposite of what you want, derive an impossible conclusion, conclude the original must be true.