Question 6

We will assume the opposite of what we want to prove.

Step 1 — Make an assumption

Let's consider two lines. Let's call them line l and line m. We are told these lines are distinct. This means they are not the same line. We want to prove they intersect at most one point. Let's assume the opposite. Let's assume they intersect at more than one point. So, they intersect at two different points. Let's call these points A and B.

$\text{Line l passes through A and B}$

$\text{Line m passes through A and B}$

$\boxed{\text{A and B are distinct points}}$

Step 2 — Find a contradiction

We have two distinct points. These are point A and point B. Line l passes through both A and B. Line m also passes through both A and B. A basic geometry rule states: "Through any two distinct points, there is exactly one line."

$\text{Line l passes through A and B}$

$\text{Line m passes through A and B}$

$\text{A and B are distinct points}$

$\text{Therefore, line l and line m must be the same line}$

$\boxed{\text{This contradicts that l and m are distinct lines}}$

Answer

(i) Our initial assumption was that two distinct lines intersect at more than one point. (ii) This assumption led to a contradiction with a basic geometry rule. (iii) Therefore, two distinct lines in a plane cannot intersect in more than one point.