Hypotheses, Conclusions, and If-Then Form | Geometry | LevelKite

You already know how to test a conjecture and how to use a counterexample to knock one down. Now you will use those skills on the kind of sentence that geometry is built from.

A conditional statement, or conditional, is a statement that can be written as "If $p$ , then $q$ ," where $p$ and $q$ are smaller statements. It claims that whenever $p$ is true, $q$ is true too. The part after if is the hypothesis, the condition you start from. The part after then is the conclusion, the result the statement claims.

\text{If } \underbrace{p}_{\text{hypothesis}}, \text{ then } \underbrace{q}_{\text{conclusion}}.

For example, take this conditional: "If an angle is a right angle, then its measure is $90^\circ$ ." A right angle is an angle whose measure is exactly $90^\circ$ .

Hypothesis: an angle is a right angle.
Conclusion: its measure is $90^\circ$ .

The words if and then are signposts. They point to the two parts, but they are not part of either one. In symbols, you write the conditional as $p \to q$ and read it as "if $p$ , then $q$ ."

Worked example

Find the two parts

Congruent angles are angles with equal measures. Identify the hypothesis and conclusion of this conditional: "Two angles are congruent if their measures are equal."

Find the word if. The part after it is the hypothesis, even when it comes last in the sentence. This part is about the measures of the two angles, so name the angles to make it stand alone.
$\text{hypothesis: two angles have equal measures}$
The rest of the sentence is what is claimed, so it is the conclusion.
$\text{conclusion: the two angles are congruent}$
Check by putting the parts in standard order. The sentence still says the same thing, so the parts are in the right places.
$\text{If two angles have equal measures, then they are congruent.}$

Answer

The hypothesis is "two angles have equal measures," and the conclusion is "the two angles are congruent."

Many facts in geometry, including theorems (statements that have been proved), are not written with if and then. Putting a sentence in if-then form means rewriting it as "If $p$ , then $q$ " without changing what it claims. This form makes it clear what you start from and what must follow. To rewrite a sentence:

Find the condition it starts from. That is the hypothesis. Words such as all, every, when, and whenever often point to it.
Find what it claims about that condition. That is the conclusion.
Write each part as a complete clause and put the clauses in the form.

Worked example

Rewrite a sentence in if-then form

Write this statement in if-then form: "All right angles are congruent."

Find the condition the sentence starts from. It speaks about right angles, so being right angles goes in the hypothesis. Congruence compares a pair of angles, so write this part about two angles.
$p:\ \text{two angles are right angles}$
Find the claim. The sentence says those angles are congruent, so that goes in the conclusion. The word they refers to the same two angles.
$q:\ \text{they are congruent}$
Put the hypothesis after if and the conclusion after then.
$\text{If two angles are right angles, then they are congruent.}$
Check the meaning against the original. Both sentences start from right angles and claim congruence. The if-then form applies to every pair of right angles, so it keeps the meaning of the word all.

Answer

In if-then form, the statement reads: "If two angles are right angles, then they are congruent."

The Vertical Angles Theorem, the Triangle Angle Sum Theorem, and an algebra statement in if-then form

Original sentence	Hypothesis	Conclusion	If-then form
Vertical angles, the opposite angles formed by two intersecting lines, are congruent.	two angles are vertical angles	they are congruent	If two angles are vertical angles, then they are congruent.
The measures of the three angles of a triangle add to $180^\circ$ .	a figure is a triangle	the measures of its three angles add to $180^\circ$	If a figure is a triangle, then the measures of its three angles add to $180^\circ$ .
$x = 4$ whenever $2x + 3 = 11$ .	$2x + 3 = 11$	$x = 4$	If $2x + 3 = 11$ , then $x = 4$ .

Now for truth. The truth value of a statement is whether it is true or false. To see what decides the truth value of a conditional, think of it as a promise. Suppose your teacher says, "If you score 90 or higher on the quiz, then you earn a bonus point." The table checks all four possible outcomes, and only one of them breaks the promise. In the last two rows, you scored below 90, so the promise does not apply to you, and nothing that happens can break it. A conditional that is not broken counts as true.

Hypothesis: you score 90 or higher	Conclusion: you earn a bonus point	Truth value of the conditional	Why
True	True	True	You met the condition and got the result, so the promise is kept.
True	False	False	You met the condition but did not get the result, so the promise is broken.
False	True	True	The promise says nothing about scores below 90, so getting a bonus point does not break it.
False	False	True	The promise says nothing about scores below 90, so missing the bonus point does not break it.

Key concept

When a conditional is false

A conditional $p \to q$ is false only when its hypothesis $p$ is true and its conclusion $q$ is false. In every other case, it is true. A true hypothesis with a false conclusion is the only way to break a conditional.

p \to q \text{ is false only when } p \text{ is true and } q \text{ is false}

Most conditionals in geometry cover many cases at once, such as every pair of angles. Use the key idea on each case:

To show a conditional is false, find one case with a true hypothesis and a false conclusion. That case is a counterexample, and one is enough.
To show a conditional is true, start from the hypothesis and reach the conclusion using definitions, postulates (statements accepted without proof), and theorems. Then no counterexample can exist.

A conditional you have not settled yet is a conjecture, so the rule you already know still applies: one counterexample disproves it, and any number of confirming examples never proves it.

Worked example

Test a conditional with a counterexample

Determine the truth value of this conditional: "If two angles are congruent, then they are right angles."

Name the parts. The hypothesis is that two angles are congruent, and the conclusion is that they are right angles. The conditional is false only if a case makes the hypothesis true and the conclusion false, so look for congruent angles that are not right angles.
Try angles A and B that each measure 40 degrees. Their measures are equal, so they are congruent by the definition of congruent angles. The hypothesis is true for this pair.
$m\angle A = m\angle B = 40^\circ$
By the definition of a right angle, a right angle measures exactly 90 degrees. These angles measure 40 degrees, so they are not right angles. The conclusion is false for this pair.
$40^\circ \neq 90^\circ$
This pair has a true hypothesis and a false conclusion, so it is a counterexample. One counterexample is enough to show that the whole conditional is false.

Answer

The conditional is false, and two angles that each measure $40^\circ$ form a counterexample.

Compare this with the statement you wrote earlier: "If two angles are right angles, then they are congruent." Each angle measures $90^\circ$ by the definition of a right angle, so the measures are equal, and the angles are congruent by the definition of congruent angles. That conditional is true, and it runs from right angles to congruent angles. The conditional you just tested swaps those parts, and it is false.

Swapping the hypothesis and conclusion makes a new statement whose truth value may differ from the original. So when you use a fact, check which direction it states, and never assume the swapped statement is true.

Worked example

Reason from the hypothesis

Determine the truth value of this conditional: "If $\angle A$ is a right angle and $m\angle A = (2x + 10)^\circ$ , then $x = 40$ ."

Separate the parts. Given (the hypothesis): angle A is a right angle and its measure is (2x + 10) degrees. To be proved (the conclusion): x = 40.
By the definition of a right angle, angle A measures 90 degrees. Both expressions name the same measure, so set them equal, by the Substitution Property of Equality.
$m\angle A = 90^\circ \text{ and } m\angle A = (2x + 10)^\circ, \text{ so } 2x + 10 = 90$
Subtract 10 from both sides, by the Subtraction Property of Equality.
$2x + 10 - 10 = 90 - 10, \text{ so } 2x = 80$
Divide both sides by 2, by the Division Property of Equality.
$\frac{2x}{2} = \frac{80}{2}, \text{ so } x = 40$
Check x = 40 against the geometry. The measure comes out to 90 degrees, which is between 0 and 180 degrees and is exactly the measure of a right angle.
$2(40) + 10 = 80 + 10 = 90$
The steps used only the hypothesis and named reasons, so they hold in every case where the hypothesis is true. No case can have a true hypothesis and a false conclusion, so no counterexample exists.

Answer

The conditional is true: whenever its hypothesis is true, the definition of a right angle and the properties of equality force $x = 40$ .

Takeaway: A conditional joins a hypothesis to a conclusion, and if-then form shows which is which. It is false only when its hypothesis is true and its conclusion is false. So one counterexample disproves it, while any number of confirming examples never proves it. To show it is true, you reason from definitions, postulates, and theorems.

Next, you will write the converse, inverse, and contrapositive of a conditional, determine their truth values, and write and evaluate biconditional statements.