Converse, Inverse, Contrapositive & Biconditionals

Q: Why does the contrapositive always match the original statement?

If $p$ guarantees $q$, then whenever $q$ fails, $p$ must have failed too — otherwise $q$ would have happened. That is exactly what the contrapositive says, so the two statements are logically equivalent.

Every conditional statement “if $p$ , then $q$ ” comes with three relatives. The converse swaps the two parts. The inverse negates both parts. The contrapositive does both — it swaps and negates. Each one is a different statement, and each can be true or false on its own.

There is one relationship worth memorizing above all the others: a conditional and its contrapositive always have the same truth value. That single fact powers a lot of proof work, so this lesson focuses on building each statement correctly and knowing which ones travel together.

The three related conditionals

Start with “if $p$ , then $q$ .” The converse is “if $q$ , then $p$ ” — swap only. The inverse is “if not $p$ , then not $q$ ” — negate only. The contrapositive is “if not $q$ , then not $p$ ” — swap and negate.

A clean way to keep them straight: the converse changes the order, the inverse changes the signs, and the contrapositive changes both. If you only change one thing, you have not built the contrapositive. The table below lines up all four forms.

Statement	Form
Conditional	If $p$ , then $q$
Converse	If $q$ , then $p$
Inverse	If not $p$ , then not $q$
Contrapositive	If not $q$ , then not $p$

Which statements share a truth value

A conditional and its contrapositive always match — both true or both false. Likewise, the converse and the inverse always match each other, because the inverse is the contrapositive of the converse.

But the original statement tells you nothing about its converse. “If a figure is a triangle, then it has three sides” is true, and its converse “if a figure has three sides, then it is a triangle” happens to be true too — while “if a figure is a square, then it has four sides” is true and its converse is false. Each direction must be checked separately.

Biconditionals

A biconditional “ $p$ if and only if $q$ ” is true only when both the conditional and its converse are true. It says the two conditions come as a package: each one guarantees the other.

Good definitions are biconditionals. “An angle is straight if and only if it measures $180^\circ$ ” works because both directions hold. “A figure is a square if and only if it has four sides” fails, because the converse direction is false.

Worked examples

Example 1: write all three relatives

Write the converse, inverse, and contrapositive of “If an angle is straight, then it measures $180^\circ$ .”

Converse — swap: “If an angle measures

180^\circ

, then it is straight.”

Inverse — negate both: “If an angle is not straight, then it does not measure

180^\circ

.”

Contrapositive — swap and negate: “If an angle does not measure

180^\circ

, then it is not straight.”

Answer: All three statements above; here every one of them is true.

Example 2: use the contrapositive relationship

“If a figure is a square, then it has four sides” is true. What can you conclude about its contrapositive, converse, and inverse?

The contrapositive always matches the original, so “if a figure does not have four sides, then it is not a square” is true

The converse must be tested separately: “if a figure has four sides, then it is a square” is false — a rectangle is a counterexample

The inverse matches the converse, so it is false too

Answer: Contrapositive: true. Converse and inverse: false.

Example 3: test a biconditional

Can “If two angles are vertical, then they are congruent” be written as a true biconditional?

The conditional itself is true — vertical angles are congruent

Check the converse: “If two angles are congruent, then they are vertical” — false, since two

40^\circ

angles in different figures are congruent but not vertical

A biconditional requires both directions to be true

Answer: No — the converse is false, so the biconditional is false.

Try one yourself

Part	Original	Contrapositive
Hypothesis	figure is a triangle	figure does not have three sides
Conclusion	figure has three sides	figure is not a triangle

Common questions

What is the difference between the inverse and the converse?

The converse swaps the hypothesis and conclusion. The inverse keeps them in place but negates both. They are different statements, though they always share the same truth value as each other.

Why does the contrapositive always match the original statement?

If $p$ guarantees $q$ , then whenever $q$ fails, $p$ must have failed too — otherwise $q$ would have happened. That is exactly what the contrapositive says, so the two statements are logically equivalent.

When is a biconditional true?

Only when the conditional and its converse are both true. If either direction has a counterexample, the biconditional is false.

How do I negate a statement like “it has three sides”?

Add “not” to the claim itself: “it does not have three sides.” Be careful not to negate into an opposite that says more than “not” — the negation of “greater than 5” is “not greater than 5,” which includes equal to 5.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1