Question

How many points are there in the intersection of two distinct lines?

Answer 1

Hint: For solving we will make use of the one of Euclid’s axioms from his postulates and try to understand how many points there in the intersection of two distinct lines are.

Complete step-by-step answer:
Given:
There are two distinct lines and we have to tell how many intersection points are there.
Now, before we proceed we should know that according to Euclid’s postulate he stated that “A straight line may be drawn from anyone point to any other point”. And from this statement, we can conclude that from two distinct points there is only one unique line that will pass through them. This is also called an axiom of Euclid’s statement. We will use this axiom to prove our result.
Now, we can say that if the lines are not parallel then, they will intersect at one unique point and we can prove it easily.
Now, let there be two distinct non-parallel lines $l$ and $m$ . We have to prove that they will have only one point in common. We will use a contradiction method to prove this. So, for the time being, let us suppose that the two distinct non-parallel lines intersect at two different points, say A and B. So, in other words, we can say that we have two different lines passing through two points A and B. But this assumption clashes with the axiom that only one unique line will pass through two distinct points. Thus, there is a very strong contradiction on our assumption and finally, we conclude that two distinct non-parallel lines will intersect at only one unique point. For more clarity look at the figure given below:

In the above figure, two non-parallel lines AB and CD intersect at a unique point E.
If the lines don’t intersect then they will be parallel or maybe skew lines. For more clarity look at the figure given below:

In the above figure, there are two parallel lines AB and CD and they don’t have any common points.

In the above figure, there are two lines AB and CD which are in different planes and they don’t have any common points.

Note: Although the question is very easy, the student should know the correct reason behind the answer. We can also think of such questions with practical examples that will help us in strengthening our concept of geometry.