Key takeaway: The converse of an if-then statement swaps the hypothesis and the conclusion, and its truth must be checked separately because it can be false even when the original statement is true.
Grade 10 – GeometryG-CO.C.9Geometry and Measurement
Definition of Converse
The converse of an if-then claim is built by exchanging its two parts. If the original says “if \(p\), then \(q\),” the new claim says “if \(q\), then \(p\).”
Some geometry theorems have a true twin, which is how equal angles can prove two lines parallel. Many do not. A true original gives no guarantee about the swapped version, so test it separately and hunt for a counterexample.
Formula and notation
If the original is \(p \to q\), the swapped version is:
\[ q \to p \]
If both directions are true, the pair can be written with a double arrow, \(p rightarrow q\).
Key facts
- Only the order of the two parts changes; neither part is negated.
- A true statement can have a false converse.
- When both directions hold, the two can be fused into a single two-way claim.
- Its truth value always matches that of the inverse of the original.
Converse example
The statement “If a quadrilateral is a rectangle, then its diagonals are congruent” is true. State the swapped version and decide whether it is true.
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The hypothesis is “a quadrilateral is a rectangle”; the conclusion is “its diagonals are congruent.”
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Exchange them: “If the diagonals of a quadrilateral are congruent, then it is a rectangle.”
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Try an isosceles trapezoid with bases 12 ft and 20 ft: its diagonals are equal, yet it has no right angles.
The swapped version is false; the isosceles trapezoid is a counterexample.
Common mistake
Watch out
Marco takes “If an animal is a poodle, then it is a dog” and writes the converse as “If an animal is not a poodle, then it is not a dog.”
Why it is wrong
Putting “not” in both parts produces the inverse. Swapping gives “If an animal is a dog, then it is a poodle,” which fails for a beagle.
Related terms and lessons
- Conditional Statement
- Two-Column Proof
- Parallel Lines
- Isosceles Triangle Theorem
- Perpendicular Bisector
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