16
Proof by contradiction
Textbook: pp. 40–41
GoalUnderstand proof by contradiction.
New words
assume the opposite · teskarisini faraz qilishcontradiction · ziddiyatmethod of proof · isbot usulinon-intersection · kesishmaslik
Explanation
In proof by contradiction we accept the hypothesis of the theorem and assume the conclusion is false. By logical steps we then reach a result that contradicts a known axiom or theorem. The contradiction shows the assumption was wrong, so the conclusion is true. Three things matter: write the opposite of the statement correctly; draw correct conclusions from the assumption; arrive at a result contradicting a known property. Theorem: two lines perpendicular to the same line do not intersect. In the proof we assume they intersect, which leads to two lines passing through two points, contradicting the axiom.
Worked examples
Claim: two different lines cannot have two common points. Assume A and B are common; then two lines pass through A and B, contradicting the axiom. So the claim is true.
Claim: a segment has only one midpoint. Assume C ≠ D are midpoints. Then AC = AD = AB : 2, but on a ray only one segment of a given length can be laid off, so C = D. Contradiction.
Class activity
“Find the contradiction”: in pairs, pupils assume the opposite of a simple statement and find the contradiction.
Practice
1
What is the opposite of “a segment has one midpoint”?
A segment has at least two midpoints.
2
In proof by contradiction, what does the contradiction show?
The assumption is false, so the statement to be proved is true.
3
In a proof by contradiction two lines ended up with two common points. By the axiom, how many lines pass through two points?
1
4
Why is “two lines through two points” a contradiction?
By the axiom only one line passes through two points.