i need proof for threom 1 -Two distinct lines cannot have more than one point in common 

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

Proof:

Let 'l' and 'm' are two distinct lines that is l ¹ m.

To Prove:  'l' and 'm' cannot have more than one common point.

Let us assume that 'l' and 'm' have two common points P and Q contrary to conclusion.
Then P and Q determine the line 'l' and also the line 'm'.
But  it is known that, there exists one and only one line containing two given points.
So 'l' and 'm' cannot be distinct.
This contradicts the supposition that l and mare distinct.
So, our assumption that 'l' and 'm' have more than one point in common is false.

 

Hence, we conclude that 'l' and 'm' cannot have more than one point in common.