Counterexamples and Testing Conjectures | Geometry | LevelKite

You have already made conjectures from patterns. A conjecture is a statement you believe because of a pattern but have not yet proved, and a pattern can mislead you. In this lesson, you learn how to test a conjecture and decide whether it is true or false.

Start by writing the conjecture as a conditional, a statement of the form "if $p$ , then $q$ ." The "if" part, $p$ , is the hypothesis. The "then" part, $q$ , is the conclusion. A conditional conjecture makes a promise about every case: whenever the hypothesis is true, the conclusion is true too.

That promise can break in only one way: a conditional is false only when its hypothesis is true and its conclusion is false. A counterexample is one specific case that does exactly this. One counterexample disproves a conjecture, because the conjecture promised every case and this case breaks the promise.

To find a counterexample, ask how the conclusion could fail. Then see whether the hypothesis can still be true in that same case.

Worked example

Break a conjecture about angles

Find a counterexample to this conjecture: "If two angles are supplementary, then one of them is obtuse."

Name the two parts. The hypothesis is that two angles are supplementary. The conclusion is that one of them is obtuse. A counterexample must make the hypothesis true and the conclusion false.
Start from a false conclusion: neither angle is obtuse, so each measure is at most 90 degrees. If either measure were below 90 degrees, the sum would fall below 180 degrees and the hypothesis would fail. So try two angles of 90 degrees each. That measure is valid because it is between 0 and 180 degrees.
$m\angle A = 90^\circ, \quad m\angle B = 90^\circ$
Check the hypothesis with the definition of supplementary angles, which says the measures add to 180 degrees. They do, so the hypothesis is true.
$m\angle A + m\angle B = 90^\circ + 90^\circ = 180^\circ$
Check the conclusion with the definition of an obtuse angle, which says the measure is greater than 90 degrees. A measure of exactly 90 degrees is not greater than 90 degrees, so neither angle is obtuse and the conclusion is false.
$90^\circ \text{ is not greater than } 90^\circ$

Answer

Two right angles, each measuring $90^\circ$ , form a counterexample, so the conjecture is false.

Not every case tests a conjecture. A confirming example makes the hypothesis and the conclusion both true. It agrees with the conjecture, but it says nothing about other cases. A case with a false hypothesis does not test the promise at all, because a conditional is false only when its hypothesis is true and its conclusion is false. The table sorts four cases for the conjecture you just disproved.

Four cases for the conjecture: if two angles are supplementary, then one of them is obtuse

Angle measures	Hypothesis: supplementary?	Conclusion: one is obtuse?	What the case shows
$120^\circ$ and $60^\circ$	True: the sum is $180^\circ$	True: $120^\circ$ is obtuse	Confirming example: agrees, but proves nothing about other cases
$90^\circ$ and $90^\circ$	True: the sum is $180^\circ$	False: neither angle is obtuse	Counterexample: the conjecture is false
$120^\circ$ and $30^\circ$	False: the sum is $150^\circ$	True: $120^\circ$ is obtuse	Not a test: the hypothesis is false, so it is not a counterexample
$30^\circ$ and $40^\circ$	False: the sum is $70^\circ$	False: neither angle is obtuse	Not a test: the hypothesis is false, so it is not a counterexample

Key concept

One counterexample disproves; examples never prove

A conditional is false only when its hypothesis is true and its conclusion is false. So one real case with a true hypothesis and a false conclusion, a counterexample, disproves a conjecture. Any number of confirming examples never proves it, because examples check only the cases you tried. To show that a conjecture is true, you need a reason that covers every case, built from definitions, postulates (statements accepted without proof) and theorems (statements already proved).

A counterexample must also be a case that can really exist. Before you use one, check the geometric conditions. Side lengths must be positive. Angle measures must be between $0^\circ$ and $180^\circ$ . The side lengths of a triangle must satisfy the Triangle Inequality Theorem: the sum of any two side lengths is greater than the third. A case that cannot exist never makes the hypothesis true, so it cannot be a counterexample.

Worked example

A counterexample that must really exist

Find a counterexample to this conjecture: "If a triangle is isosceles, then it is equilateral." A triangle is isosceles when at least two of its sides are congruent, and equilateral when all three sides are congruent. Sides are congruent when their lengths are equal.

Build the case. Two sides of equal length make the hypothesis true, and a third side of a different length makes the conclusion false. Try side lengths 5, 5 and 8, which are all positive.
Check that this triangle can exist, using the Triangle Inequality Theorem. The two different sums of two side lengths are each greater than the remaining side. Lengths such as 2, 2 and 8 would fail, because 2 + 2 = 4 is less than 8.
$5 + 5 = 10 > 8, \quad 5 + 8 = 13 > 5$
Check the hypothesis with the definition of an isosceles triangle. Two sides have equal lengths, so they are congruent and the hypothesis is true.
$5 = 5$
Check the conclusion with the definition of an equilateral triangle. The third side has a different length, so the three sides are not all congruent and the conclusion is false.
$8 \neq 5$

Answer

A triangle with side lengths $5$ , $5$ and $8$ is a counterexample, so the conjecture is false.

Now you can decide whether a conjecture is true or false. Work in this order:

Write it as a conditional. Name the hypothesis, which is what you are given, and the conclusion, which is what you must prove or break.
Look for a counterexample: a real case with a true hypothesis and a false conclusion. Check each part from stated measures and definitions, never from how a sketch looks.
If you find one, the conjecture is false. One counterexample is enough.
If you do not find one, the conjecture is not proved yet. Failing to find a counterexample is no more a proof than a list of confirming examples. Look for a reason that starts from the hypothesis and reaches the conclusion in every case, using definitions, postulates and theorems, and never the conclusion itself. When you have that reason, the conjecture is true.

Worked example

Decide a conjecture about supplementary angles

Decide whether this conjecture is true or false: "If two angles are each supplementary to the same angle, then they are congruent." Angles are congruent when their measures are equal.

Hunt for a counterexample. Let angle C measure 110 degrees. By the definition of supplementary angles, an angle supplementary to it must measure 70 degrees, a valid measure between 0 and 180 degrees, so the two angles match. This confirming example proves nothing by itself, but it hints at why no counterexample appears: the measure of angle C fixes both angles.
$180^\circ - 110^\circ = 70^\circ$
Separate what is given from what is to be proved. Given, from the hypothesis: angle A and angle C are supplementary, and angle B and angle C are supplementary. To prove, from the conclusion: angle A is congruent to angle B. Using letters instead of numbers lets the reason cover every case.
Apply the definition of supplementary angles to each given pair, reading it from "supplementary" to "the measures add to 180 degrees."
$m\angle A + m\angle C = 180^\circ, \quad m\angle B + m\angle C = 180^\circ$
Both sums equal 180 degrees, so they equal each other by the Transitive Property of Equality.
$m\angle A + m\angle C = m\angle B + m\angle C$
Subtract the measure of angle C from both sides, which keeps the equation true by the Subtraction Property of Equality.
$m\angle A = m\angle B$
Read the definition of congruent angles from "equal measures" to "congruent." It gives the conclusion you set out to prove.
$\angle A \cong \angle B$

Answer

The conjecture is true: for every angle $C$ , two angles supplementary to it are congruent. Now that it is proved, this statement is a theorem, known as the Congruent Supplements Theorem.

Every conditional comes with three related statements. To negate a statement is to state its opposite, such as "not $p$ ." Starting from "if $p$ , then $q$ ," the converse swaps the two parts, the inverse negates both parts, and the contrapositive swaps and negates them.

A statement's truth value is whether it is true or false. The table below applies one rule to each form: a conditional is false only when its hypothesis is true and its conclusion is false. A conditional and its contrapositive fail in exactly the same case, so they always share a truth value. The converse and the inverse fail in exactly the same case as each other, so they share a truth value too. That is a different case, so their truth value may differ from the original's.

So a true statement does not make its converse true. A definition can be read in both directions, but a theorem works only in the direction it states unless its converse is also proved. Always say which direction you are using.

When each form of a conditional is false

Form	Statement	False only when	Always shares its truth value with
Original	If $p$ , then $q$ .	$p$ is true and $q$ is false	The contrapositive
Contrapositive	If not $q$ , then not $p$ .	$q$ is false and $p$ is true (the same case as the original)	The original
Converse	If $q$ , then $p$ .	$q$ is true and $p$ is false	The inverse
Inverse	If not $p$ , then not $q$ .	$p$ is false and $q$ is true (the same case as the converse)	The converse

Worked example

Decide a conjecture and its related statements

Use these definitions: a quadrilateral is a closed figure with four straight sides, a rectangle is a quadrilateral with four right angles, and a square is a quadrilateral with four right angles and four congruent sides. Decide whether "If a quadrilateral is a square, then it is a rectangle" is true or false. Then do the same for its contrapositive, converse and inverse.

Original, from square to rectangle. By the definition of a square, any square has four right angles. By the definition of a rectangle, read from four right angles to rectangle, it is a rectangle. This reason covers every square, so the original is true.
$\text{square} \Rightarrow \text{four right angles} \Rightarrow \text{rectangle}$
Contrapositive: "If a quadrilateral is not a rectangle, then it is not a square." A conditional and its contrapositive always share a truth value, so this statement is true.
Converse, from rectangle to square: "If a quadrilateral is a rectangle, then it is a square." Take a quadrilateral with four right angles and side lengths 3, 5, 3 and 5. By the definition of a rectangle, the hypothesis is true. Its sides are not all congruent, so by the definition of a square, the conclusion is false. This counterexample makes the converse false.
$3 \neq 5$
Inverse: "If a quadrilateral is not a square, then it is not a rectangle." The converse and the inverse share a truth value, so this statement is false too. The same figure shows it: it is not a square, so the hypothesis is true, and it is a rectangle, so the conclusion is false.

Answer

The original and its contrapositive are true. The converse and the inverse are false, and the rectangle with side lengths $3$ , $5$ , $3$ and $5$ is a counterexample to both.

To test a conjecture, look for a real case with a true hypothesis and a false conclusion, the only case that makes a conditional false. One counterexample disproves the conjecture, and any number of confirming examples never proves it, so calling a conjecture true takes a reason that covers every case. A conditional and its contrapositive always share a truth value, while the converse and the inverse share one that may differ from the original.