Key takeaway: A conditional statement is an if-then sentence that links a hypothesis to a conclusion, and it is false only when the hypothesis is true while the conclusion is false.
Grade 10 – GeometryG-CO.C.9Geometry and Measurement
Definition of Conditional Statement
A conditional statement claims that one fact forces another. It reads “if \(p\), then \(q\).” The “if” clause is the hypothesis, and the “then” clause is the conclusion.
To disprove it you need a counterexample, a single case where the hypothesis is satisfied but the conclusion fails. To prove it, an argument has to cover every possible case. Rewriting a plain sentence into if-then form is a good habit, because it separates what is assumed from what is claimed.
Formula and notation
A conditional statement is written with an arrow that reads “implies”:
\[ p \to q \]
Reversing the parts gives \(q \to p\). Negating both parts gives \(\sim p \to \sim q\). Reversing and negating gives \(\sim q \to \sim p\). The tilde means “not.”
Key facts
- A conditional statement fails is true hypothesis with false conclusion.
- A statement and its reversed-and-negated form are true or false together.
- Reversing the parts, or negating both, can turn a true claim into a false one.
- One counterexample is enough to settle that a claim is false.
Conditional Statement example
Start from the true statement “If a polygon is a regular hexagon, then each interior angle measures \(120^{\circ}\).” Form the reversed, double-negated and reversed-negated versions, and judge each.
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Reversed: “If each interior angle of a polygon measures \(120^{\circ}\), then it is a regular hexagon.” False: a hexagon with sides 3, 5, 3, 5, 3, 5 has all angles \(120^{\circ}\) but unequal sides.
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Both negated: “If a polygon is not a regular hexagon, then its angles are not all \(120^{\circ}\).” False, by the same hexagon.
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Reversed and negated: “If the angles of a polygon are not all \(120^{\circ}\), then it is not a regular hexagon.” True, since a regular hexagon always has \(120^{\circ}\) angles.
Only the original and the reversed-negated form are true; the other two are false.
Common mistake
Watch out
Zara says “If a number ends in 0, then it is divisible by 5” is false, because 15 is divisible by 5 and does not end in 0.
Why it is wrong
A counterexample must make the hypothesis true and the conclusion false. The number 15 fails the hypothesis, so it says nothing about this claim, which is true. Her example only disproves the reversed claim.
Related terms and lessons
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