Math Core

Almost every rule in geometry is written as an "if-then" statement: if two angles are vertical, then they are congruent. Learning to read, rewrite and test these statements is the first real skill of proof-writing, because a proof is just a chain of if-then steps.

Hypothesis and conclusion

A conditional statement has the form "if $p$ , then $q$ ," written $p \to q$ .

The hypothesis $p$ is the part after "if."
The conclusion $q$ is the part after "then."

In "If an angle is acute, then its measure is less than $90^\circ$ ," the hypothesis is "an angle is acute" and the conclusion is "its measure is less than $90^\circ$ ."

Many statements hide their if-then form. "All squares are rectangles" means "If a figure is a square, then it is a rectangle." "Vertical angles are congruent" means "If two angles are vertical angles, then they are congruent."

When is a conditional true?

A conditional makes a promise: whenever the hypothesis happens, the conclusion happens too. It is false only when the promise is broken, meaning the hypothesis is true but the conclusion is false. So to show a conditional is false, give a counterexample where the hypothesis holds and the conclusion fails.

"If a number is prime, then it is odd" is false: $2$ is prime (hypothesis true) but not odd (conclusion false).

Related conditionals

Starting from $p \to q$ , you can build three new statements by swapping and negating the parts. The negation of $p$ , written $\sim p$ , is "not $p$ ."

Definition

Converse, inverse and contrapositive

statement	form	in words
conditional	$p \to q$	if $p$ , then $q$
converse	$q \to p$	if $q$ , then $p$ (swap)
inverse	$\sim p \to \sim q$	if not $p$ , then not $q$ (negate)
contrapositive	$\sim q \to \sim p$	if not $q$ , then not $p$ (swap and negate)

Take the true statement "If an animal is a fish, then it can swim."

Converse: If an animal can swim, then it is a fish. False (a dog can swim).
Inverse: If an animal is not a fish, then it cannot swim. False (the dog again).
Contrapositive: If an animal cannot swim, then it is not a fish. True.

That pattern is no accident.

Equivalent statements

A conditional and its contrapositive always have the same truth value.

The converse and the inverse always have the same truth value as each other (they are contrapositives of each other), but that value may differ from the original.

Why does the contrapositive match? If every fish can swim, then anything that can't swim can't be a fish. Both statements describe exactly the same situation from different directions.

Common mistake

The most common mistake in logic is assuming the converse is true because the original is. "If two angles are right angles, then they are congruent" is true. Its converse, "If two angles are congruent, then they are right angles," is false: two $40^\circ$ angles are congruent. Always test the converse separately.

Biconditionals and definitions

When a conditional and its converse are both true, you can combine them into a biconditional using "if and only if," written $p \leftrightarrow q$ .

"If an angle is a right angle, then it measures $90^\circ$ " and "If an angle measures $90^\circ$ , then it is a right angle" are both true, so: "An angle is a right angle if and only if it measures $90^\circ$ ."

Every good definition can be written as a biconditional, because a definition works in both directions. The definition of a midpoint says: a point is the midpoint of a segment if and only if it divides the segment into two congruent segments. That's why, in a proof, you can use a definition forward (" $M$ is the midpoint, so $\overline{AM} \cong \overline{MB}$ ") or backward (" $\overline{AM} \cong \overline{MB}$ with $M$ on $\overline{AB}$ , so $M$ is the midpoint").

Worked example: Rewriting in if-then form

Write "Complementary angles have measures that add to $90^\circ$ " as a conditional. Identify the hypothesis and conclusion.

Solution. If two angles are complementary, then their measures add to $90^\circ$ . Hypothesis: two angles are complementary. Conclusion: their measures add to $90^\circ$ .

Worked example: All four statements

Write the converse, inverse and contrapositive of "If $x = 5$ , then $x^2 = 25$ ," and decide whether each is true.

Solution.

Conditional: If $x = 5$ , then $x^2 = 25$ . True.
Converse: If $x^2 = 25$ , then $x = 5$ . False: $x = -5$ is a counterexample.
Inverse: If $x \ne 5$ , then $x^2 \ne 25$ . False: again $x = -5$ .
Contrapositive: If $x^2 \ne 25$ , then $x \ne 5$ . True, matching the original.

Worked example: Can it be a biconditional?

Can "If two angles form a linear pair, then they are supplementary" be written as a true biconditional?

Solution. The statement is true. Check the converse: "If two angles are supplementary, then they form a linear pair." A $100^\circ$ angle in one corner of the page and an $80^\circ$ angle in another are supplementary but not adjacent, so they don't form a linear pair. The converse is false, so no true biconditional is possible.

Tip

To test a biconditional, test both directions. One counterexample in either direction makes the whole biconditional false.

Practice

Practice 1

What is the hypothesis of "If a triangle has three congruent sides, then it is equilateral"?

Practice 2

What is the converse of "If it is raining, then the ground is wet"?

Practice 3

What is the contrapositive of "If $m\angle A = 30^\circ$ , then $\angle A$ is acute"?

Practice 4

A conditional statement is true. Which related statement must also be true?

Practice 5

Which value of $x$ is a counterexample to "If $x^2 > 16$ , then $x > 4$ "?

Practice 6

Which statement can be written as a true biconditional?

Practice 7

The statement "If a quadrilateral is a square, then it has four right angles" is true. Its converse is false. What are the truth values of its inverse and contrapositive?