Question 26 of 26intermediate🔧 ApplyLong Answer3 marks

Prove by contradiction that two distinct lines in a plane cannot intersect in more than one point.

Correct Answer

Given: Two distinct lines in a plane.

To Prove: Two distinct lines cannot intersect in more than one point.

Proof by Contradiction:

Let us assume that two distinct lines can intersect in more than one point. Let the two distinct lines be ll and mm, and let them intersect at two distinct points AA and BB.

Since both lines ll and mm pass through the same two points AA and BB, and we know that through two distinct points, there passes exactly one unique line, this implies that lines ll and mm must coincide (be the same line).

But this contradicts our assumption that ll and mm are distinct lines.

Since our assumption leads to a contradiction, the assumption must be false.

∴ Two distinct lines in a plane cannot intersect in more than one point.

Exercise: EXERCISE A1.6 | Q: 6 | (Page 21)
For More Understanding

Explanation

This proof uses the method of proof by contradiction as explained in the textbook context. The key steps are:

  1. Assume the negation of what we want to prove.
  2. Show that this assumption leads to a logical contradiction.
  3. Conclude that the original statement must be true.

The contradiction arises from the fundamental axiom that through two distinct points, only one unique line can pass.

Solution Steps

  1. Step 1: Assume the negation — two distinct lines CAN intersect in more than one point.

  2. Step 2: Let lines ll and mm intersect at two points AA and BB.

  3. Step 3: Both lines pass through AA and BB, but through two points only one line can pass.

  4. Step 4: This contradicts the assumption that lines are distinct.

  5. Step 5: Therefore, two distinct lines cannot intersect in more than one point.

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