Converse
In short: A converse is formed by swapping the two halves of an if-then sentence, so “if p, then q” turns into “if q, then p”, and its truth must be checked on its own.
GeometryGeometryG-CO.C.9
What is Converse?
The converse of a conditional sentence turns it around: the old conclusion becomes the new condition, and the old condition becomes the new result. From “if p, then q” you obtain “if q, then p”.
A statement and its reverse are separate claims with separate truth values. The Pythagorean theorem is true and so is its reverse, yet “if two angles are vertical, then they are congruent” has a false reverse, since two congruent angles need not be vertical. When both directions hold, they merge into one if and only if sentence.
How to spot it
- A question says write the reverse of a statement, or asks whether a theorem also works backward.
- A proof ends with a fact that you now want to use as the starting point of a new argument.
- Phrases such as if and only if, or conditions that must hold in both directions, appear.
- A classmate claims a rule must work backward as well, and you need to test the claim.
The method
- Write the original in the form: if (hypothesis), then (conclusion).
- Exchange the two clauses, keeping each one’s wording, and smooth the grammar.
- Judge the new sentence by itself: search for a counterexample, or justify it for every case.
- If the original and the swapped sentence are both true, join them into one if and only if statement.
Converse: solved example
Problem
A sail maker states: “If a triangle is equilateral, then it is isosceles.” Write the reverse sentence and decide whether it holds.
Solution
The original is true: three equal sides include at least two equal sides. Swap the clauses: If a triangle is isosceles, then it is equilateral.
Try a triangle with sides 7, 7 and 10 inches. Two sides match, so it is isosceles, but the third side differs, so it is not equilateral. That counterexample makes the swapped claim false.
Try it
Question 1
State the reverse of “If \(x = 4\), then \(x^2 = 16\)” and say whether it is true.
Show the answer
“If \(x^2 = 16\), then \(x = 4\).” False: \(x = -4\) also gives \(x^2 = 16\).
Question 2
Write the converse of “If it is Tuesday, then the band rehearses.”
Show the answer
“If the band rehearses, then it is Tuesday.”
Converse vs. Conditional Statement
The conditional statement is the original claim: hypothesis first, conclusion second. The converse is a brand-new claim built from it by swapping those parts. It is neither a negation nor a proof of the original. “If it rains, the grass is wet” is true; its swapped form can fail, because a sprinkler also wets grass.
Keep going
Lessons
Worksheets and quizzes
Related words
- Conditional Statement
- Two-Column Proof
- Pythagorean Theorem
- Isosceles Triangle Theorem
- Parallel Lines
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