What is the difference between the converse and the inverse?

The converse swaps the hypothesis and conclusion (q arrow p); the inverse negates both without swapping (p arrow q). They are different sentences but they are logically equivalent, meaning they always have the same truth value. So if you determine one is false, the other is false too, and usually the same counterexample handles both.

Why is the contrapositive always true when the original statement is true?

Because they describe the same situation from opposite ends. "If p, then q" promises that p never occurs without q. If you find something lacking q, then p could not have been present, which is exactly "if not q, then not p." This equivalence is useful in proofs: when proving a statement directly is awkward, you can prove its contrapositive instead.

How many counterexamples do I need to disprove a statement?

Exactly one, as long as it truly satisfies the hypothesis and truly fails the conclusion. A conditional claims something about every case, so one failure breaks it. The reverse is not true — no pile of confirming examples proves a conditional, which is why geometry moves from inductive observation to formal proof.

Can a conditional statement be true if the hypothesis never happens?

Yes. A conditional is false only when the hypothesis is true and the conclusion is false. If nothing satisfies the hypothesis, that failing case can never occur, so the statement is considered true — logicians call this vacuously true. It sounds odd, but it keeps the definition consistent: the statement simply makes no promise about cases outside its hypothesis.

GEOM-2.2

Conditional Statements, Converses & Counterexamples

Learn to rewrite claims in if-then form, build converses, inverses, and contrapositives, and disprove false statements with one counterexample in Geometry 2.2.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Conditional Statements, Converses & Counterexamples, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Geometry runs on precise claims. "Vertical angles are congruent" and "If two angles are vertical angles, then they are congruent" say the same thing, but only the second one shows you exactly what you are given and exactly what you must prove. Learning to hear the hidden if-then inside an ordinary sentence is the first real step toward writing proofs.

In this lesson you will rewrite everyday geometric claims in if-then form, identify the hypothesis and conclusion, and build three related statements from any conditional: the converse, the inverse, and the contrapositive. Then comes the sharpest tool in the set — the counterexample. A single well-chosen figure can destroy a claim that sounded perfectly reasonable, and knowing how to hunt for one will keep you from proving things that are not true.

If-Then Form: Hypothesis and Conclusion

A conditional statement has the form "If

p

, then

q

," written symbolically as

p \rightarrow q

. The part after "if" is the hypothesis (

p

); the part after "then" is the conclusion (

q

). The hypothesis is what you assume; the conclusion is what follows.

Most claims in a textbook are not already in if-then form, so you have to translate. Watch for these patterns:

Original claim	If-then form
All squares are rectangles.	If a figure is a square, then it is a rectangle.
Vertical angles are congruent.	If two angles are vertical angles, then they are congruent.
A number divisible by 4 is even.	If a number is divisible by 4, then it is even.
Perpendicular lines form right angles.	If two lines are perpendicular, then they form right angles.
Only seniors may park in Lot B.	If a student parks in Lot B, then the student is a senior.

Two traps show up constantly. First, students flip "All A are B" backwards. "All squares are rectangles" becomes "If square, then rectangle" — the smaller, more specific category is the hypothesis. Second, "only if" reverses the order: "A polygon is a triangle only if it has three sides" means "If a polygon is a triangle, then it has three sides."

Also notice that a conditional makes no claim at all when the hypothesis is false. "If it is a square, then it is a rectangle" says nothing about circles. That is why counterexamples must satisfy the hypothesis — a circle cannot be used to attack this statement.

Converse, Inverse, and Contrapositive

From one conditional you can build three relatives by swapping parts, negating parts, or both. Let

\sim p

mean "not

p

Name	Form	Symbols	Built by
Conditional	If $p$ , then $q$	$p \rightarrow q$	original
Converse	If $q$ , then $p$	$q \rightarrow p$	swapping
Inverse	If not $p$ , then not $q$	$\sim p \rightarrow \sim q$	negating
Contrapositive	If not $q$ , then not $p$	$\sim q \rightarrow \sim p$	swapping and negating

Take "If a figure is a square, then it is a rectangle" (true). Converse: "If a figure is a rectangle, then it is a square" — false, since a 2-by-5 rectangle is not a square. Inverse: "If a figure is not a square, then it is not a rectangle" — also false, same 2-by-5 figure. Contrapositive: "If a figure is not a rectangle, then it is not a square" — true.

That pattern is not a coincidence. A conditional and its contrapositive are logically equivalent: they are true in exactly the same situations, always sharing a truth value. The converse and inverse are also logically equivalent to each other. So the four statements split into two pairs, and you only ever have to check two truth values, not four.

The most common mistake here is assuming the converse of a true statement is true. It often is not. This error has a name outside class too — hearing "if it rains, the game is cancelled" and concluding the game being cancelled proves it rained.

Counterexamples: Killing a Claim With One Case

A counterexample is a single case in which the hypothesis is true but the conclusion is false. One counterexample proves a conditional false, permanently. No number of supporting examples proves one true — that is the limit of inductive reasoning, and it is why geometry eventually needs proof.

To build a counterexample, work in order. First, identify the hypothesis and conclusion exactly. Second, find an object that genuinely satisfies the hypothesis. Third, check that it fails the conclusion. Fourth, state it in one clear sentence, usually with a specific figure, number, or sketch.

Suppose the claim is "If two angles are supplementary, then they are both acute." Hypothesis: the angles are supplementary. Take measures of 120 degrees and 60 degrees. They sum to 180, so the hypothesis holds, but 120 degrees is obtuse, so the conclusion fails. Done.

Where students go wrong: offering a case that does not satisfy the hypothesis. Against "If two angles are supplementary, then they are both acute," the pair 50 degrees and 40 degrees is useless — those angles are complementary, not supplementary, so the statement never applied to them in the first place.

A second common slip is a vague counterexample such as "a weird triangle." Be concrete: give side lengths, angle measures, or a labeled sketch. A third is offering a case without confirming both halves: it has to satisfy the hypothesis and it has to fail the conclusion. Check each one explicitly before you commit.

Numeric claims work the same way. "If

x^2 = 25

, then

x = 5

" dies instantly with

x = -5

When Both Directions Are True: Biconditionals and Definitions

Sometimes a conditional and its converse are both true. In that case you can combine them into a biconditional, written "

p

if and only if

q

" or

p \leftrightarrow q

.

Example: "If two angles are right angles, then they measure 90 degrees" is true, and its converse "If two angles measure 90 degrees, then they are right angles" is also true. So "An angle is a right angle if and only if it measures 90 degrees" is a valid biconditional.

This matters more than it looks. Every good definition in geometry is biconditional — it works in both directions. That is what separates a definition from a theorem. "A triangle is equilateral if and only if all three sides are congruent" is a definition, so you may use it either way in a proof. But "If a triangle is equilateral, then it is equiangular" is a theorem, and although its converse happens to be true, you must justify that separately rather than assume it.

A quick test for whether a proposed definition is any good: write its converse. If the converse is false, the definition is too broad or too narrow. "A square is a figure with four right angles" fails, because the converse admits every non-square rectangle.

When you break a biconditional apart on a homework problem, you should produce two conditionals and check each one. Writing only one direction and calling the job finished is the usual place this goes wrong.

Truth Tables and Reading Truth Values Quickly

You do not need a full logic course, but one fact about

p \rightarrow q

surprises everyone: the only way a conditional is false is when

p

is true and

q

is false. If the hypothesis is false, the whole conditional counts as true — vacuously.

$p$	$q$	$p \rightarrow q$
T	T	T
T	F	F
F	T	T
F	F	T

Read the second row carefully: that row is exactly what a counterexample produces. Finding a counterexample means finding a real instance of "hypothesis true, conclusion false."

The last two rows explain a statement like "If a triangle has four sides, then it is a circle." Nothing satisfies the hypothesis, so nothing can ever put you in row two, and the statement is technically true. Students find this strange; the point is that a conditional only promises something about cases where the hypothesis actually happens.

A practical routine for these problems: rewrite the claim in if-then form, decide whether it is true, then write the converse and decide separately. If both are true, the contrapositive and inverse are true as well and you can say so in one line. If the original is true and the converse is false, you already know the contrapositive is true and the inverse is false — and you should have a counterexample ready that handles both, since the converse and inverse fail for the same reason.

Key terms

Conditional statement.: A statement of the form "If $p$ , then $q$ " ( $p \rightarrow q$ ), where $p$ is assumed and $q$ is claimed to follow.
Hypothesis.: The "if" part of a conditional, symbolized $p$ ; the condition assumed to be true.
Conclusion.: The "then" part of a conditional, symbolized $q$ ; what the statement claims must follow.
Converse.: The statement $q \rightarrow p$ , formed by interchanging the hypothesis and conclusion. Its truth value is independent of the original.
Inverse.: The statement $\sim p \rightarrow \sim q$ , formed by negating both parts. Logically equivalent to the converse.
Contrapositive.: The statement $\sim q \rightarrow \sim p$ , formed by negating and interchanging both parts. Always has the same truth value as the original conditional.
Counterexample.: A single specific case in which the hypothesis is true and the conclusion is false, proving a conditional statement false.
Biconditional.: A statement of the form " $p$ if and only if $q$ " ( $p \leftrightarrow q$ ), true only when a conditional and its converse are both true; the form of every good definition.

Worked example

Consider the claim: "An angle measuring more than 90 degrees is obtuse." Rewrite it in if-then form, identify the hypothesis and conclusion, write the converse, inverse, and contrapositive, and determine the truth value of each. Provide a counterexample for any false statement.

Step 1 — Rewrite in if-then form. "If an angle measures more than 90 degrees, then the angle is obtuse."

Step 2 — Name the parts. Hypothesis

p

: an angle measures more than 90 degrees. Conclusion

q

: the angle is obtuse.

Step 3 — Judge the original. An obtuse angle is defined as one whose measure is between 90 and 180 degrees. Is every angle over 90 degrees obtuse? Not quite: a straight angle measures 180 degrees, and a reflex angle can measure 200 degrees. Neither is obtuse. So the conditional is false, and a counterexample is an angle measuring 180 degrees — it satisfies the hypothesis (more than 90) but fails the conclusion (a straight angle is not obtuse).

Step 4 — Converse ( $q \rightarrow p$ ). "If an angle is obtuse, then it measures more than 90 degrees." By definition every obtuse angle measures between 90 and 180 degrees, so this is true.

Step 5 — Inverse ( $\sim p \rightarrow \sim q$ ). "If an angle does not measure more than 90 degrees, then it is not obtuse." An angle of 90 degrees or less is right or acute, never obtuse, so this is true. This matches the rule that the inverse and converse always share a truth value.

Step 6 — Contrapositive ( $\sim q \rightarrow \sim p$ ). "If an angle is not obtuse, then it does not measure more than 90 degrees." The same 180-degree straight angle works again: it is not obtuse, yet it does measure more than 90 degrees. False — as expected, since the contrapositive must match the original.

Step 7 — Summarize. Original false, contrapositive false (counterexample: a 180-degree straight angle); converse true, inverse true.

Practice questions

Which statement is the contrapositive of "If a quadrilateral is a rhombus, then it has four congruent sides"?

If a quadrilateral has four congruent sides, then it is a rhombus.
If a quadrilateral is not a rhombus, then it does not have four congruent sides.
If a quadrilateral does not have four congruent sides, then it is not a rhombus.
If a quadrilateral has four congruent sides, then it is not a rhombus.

Answer: If a quadrilateral does not have four congruent sides, then it is not a rhombus.

The contrapositive is

\sim q \rightarrow \sim p

: negate both parts and swap them. Here

p

is "is a rhombus" and

q

is "has four congruent sides," so the contrapositive begins with "does not have four congruent sides" and ends with "is not a rhombus." The first choice is the converse (swap only) and the second is the inverse (negate only) — both are common mix-ups. Since the original statement is true, the contrapositive must also be true, which is a quick way to sanity-check your answer.

Write "All prime numbers greater than 2 are odd" in if-then form, then write its converse and decide whether the converse is true. Justify your answer.

Answer: If-then form: If a number is a prime number greater than 2, then it is odd. Converse: If a number is odd, then it is a prime number greater than 2. The converse is false; a counterexample is 9, which is odd but not prime (9 = 3 × 3).

For "All A are B," the specific category A becomes the hypothesis. The original conditional is true, because any even number greater than 2 has 2 as a factor and cannot be prime. But truth of a conditional says nothing about its converse. To disprove the converse you need a number that satisfies its hypothesis (odd) and fails its conclusion (prime and greater than 2). The number 9 does exactly that; 15, 21, and 25 all work as well. Notice that 4 would be a poor choice — it is not odd, so it never satisfies the hypothesis and cannot serve as a counterexample.

A student claims: "If two angles are congruent, then they are vertical angles." Decide whether the statement is true or false. If it is false, give a counterexample and then state a related conditional that is true.

Answer: False. Counterexample: the two base angles of an isosceles triangle each measure 70 degrees, so they are congruent, but they sit at two different vertices and are not formed by a pair of intersecting lines, so they are not vertical angles. The converse, "If two angles are vertical angles, then they are congruent," is true.

Start by locating the hypothesis (two angles are congruent) and the conclusion (they are vertical angles). A counterexample must satisfy the hypothesis, so pick any pair of congruent angles that clearly are not vertical — two 70-degree base angles, or the four 90-degree corners of a rectangle. Because congruence has many sources besides vertical angles, the conclusion is far too strong. Reversing the statement gives the Vertical Angles Theorem, which is true. This pair is a good reminder that a true theorem and its false converse can look almost identical on the page.

FAQ

What is the difference between the converse and the inverse?: The converse swaps the hypothesis and conclusion ( $q \rightarrow p$ ); the inverse negates both without swapping ( $\sim p \rightarrow \sim q$ ). They are different sentences but they are logically equivalent, meaning they always have the same truth value. So if you determine one is false, the other is false too, and usually the same counterexample handles both.
Why is the contrapositive always true when the original statement is true?: Because they describe the same situation from opposite ends. "If $p$ , then $q$ " promises that $p$ never occurs without $q$ . If you find something lacking $q$ , then $p$ could not have been present, which is exactly "if not $q$ , then not $p$ ." This equivalence is useful in proofs: when proving a statement directly is awkward, you can prove its contrapositive instead.
How many counterexamples do I need to disprove a statement?: Exactly one, as long as it truly satisfies the hypothesis and truly fails the conclusion. A conditional claims something about every case, so one failure breaks it. The reverse is not true — no pile of confirming examples proves a conditional, which is why geometry moves from inductive observation to formal proof.
Can a conditional statement be true if the hypothesis never happens?: Yes. A conditional is false only when the hypothesis is true and the conclusion is false. If nothing satisfies the hypothesis, that failing case can never occur, so the statement is considered true — logicians call this vacuously true. It sounds odd, but it keeps the definition consistent: the statement simply makes no promise about cases outside its hypothesis.

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