If-Then Statements

Geometry states almost every fact as an if-then sentence, called a conditional: If $p$, then $q$. The part after "if" is the hypothesis; the part after "then" is the conclusion. You already know the words point, line and plane from Points, Lines and Planes; now we look at how sentences about them fit together.

A conditional is false in only one situation: the hypothesis is true and the conclusion is false. So a single counterexample is enough to show a conditional is false. To show it is true, you need a reason that covers every case.

The four related conditionals. Converse swaps; inverse negates; contrapositive does both.
Form	Pattern	Example	True?
Statement	If $p$, then $q$.	If an angle measures $30^\circ$, then it is acute.	yes
Converse	If $q$, then $p$.	If an angle is acute, then it measures $30^\circ$.	no
Inverse	If not $p$, then not $q$.	If an angle does not measure $30^\circ$, then it is not acute.	no
Contrapositive	If not $q$, then not $p$.	If an angle is not acute, then it does not measure $30^\circ$.	yes

Why it works

Picture all the things where $p$ is true as a circle sitting inside a bigger circle of things where $q$ is true. "If $p$, then $q$" says the $p$ circle is inside the $q$ circle. Then anything outside the $q$ circle is also outside the $p$ circle: that is the contrapositive, and it must be true too. But a point can be in the $q$ circle without being in the $p$ circle, so the converse can fail.

"If $p$, then $q$": the $p$ circle sits inside the $q$ circle. A $45^\circ$ angle is acute but not $30^\circ$, so the converse fails.

Key idea

A statement and its contrapositive are always both true or both false. The converse and the inverse match each other, but not necessarily the statement. When a statement and its converse are both true, combine them into a biconditional: $p$ if and only if $q$.

Worked example: Writing the related conditionals

Write the converse, inverse and contrapositive of: If two angles are vertical angles, then they are congruent. Which are true?

Converse: If two angles are congruent, then they are vertical angles. False: two $40^\circ$ angles in different places are congruent but not vertical.
Inverse: If two angles are not vertical angles, then they are not congruent. False, by the same example.
Contrapositive: If two angles are not congruent, then they are not vertical angles. True, because the statement is true.

Answer: Only the contrapositive is true (along with the statement).

Worked example: A counterexample

Show that this is false: If $x^2 = 49$, then $x = 7$.

We need a value that makes the hypothesis true and the conclusion false.
$x = -7$: $(-7)^2 = 49$ is true, but $-7 \ne 7$.

Answer: $x = -7$ is a counterexample.

Worked example: A biconditional

Can If an angle is a right angle, then it measures $90^\circ$ be written as a biconditional?

The converse is If an angle measures $90^\circ$, then it is a right angle, which is also true.
Both directions hold, so: An angle is a right angle if and only if it measures $90^\circ$.

Answer: Yes. Every good definition works in both directions like this.

Common mistake

Assuming the converse is true because the statement is. "If a figure is a square, then it has four right angles" is true, but its converse, "If a figure has four right angles, then it is a square," is false: a $3 \times 5$ rectangle is a counterexample. Before you use a converse, look for a counterexample.

If-Then Statements

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