Theorem 5.1 - Chapter 5 Class 9 Introduction to Euclid's Geometry

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

Given : Two distinct lines l ₁   and l ₂

To Prove : l ₁   and l ₂ cannot have more than 1 point in common

Proof : We will prove this by contradiction.

Let  l ₁   & l ₂  have two points in common, P & Q.

Now, by Axiom 5.1
Given two distinct points, there is a unique line that passes through them.

Thus, only one line passes through two distinct points P & Q.

But here we assumed both l ₁   & l ₂   pass through P & Q.

So, our assumption is wrong.

Therefore,

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

Hence proved

Thus, only one line passes through two distinct points P & Q.

But here we assumed both l ₁   & l ₂   pass through P & Q.

So, our assumption is wrong.

Therefore,

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

Hence proved