Deductive Reasoning and the Law of Detachment | Geometry | LevelKite

You have already looked at geometric patterns and made conjectures, unproved statements that you think are true. In this lesson, you will see the difference between reasoning that suggests a statement and reasoning that proves it. Then you will use a rule that gives a conclusion that must be true whenever its two conditions are met.

Inductive reasoning goes from specific examples to a general conjecture. You look at some cases, notice a pattern, and predict that the pattern always holds. This is a great way to find ideas, but it cannot prove them. Any number of confirming examples never proves a conjecture, and one counterexample, a single case where the statement is false, disproves it.

Here is a pattern that fails. Mark points on a circle and join every pair with a segment, placing the points so that no three segments meet at one point inside the circle. Then count the regions inside the circle. With $2$ , $3$ , $4$ and $5$ points, you get $2$ , $4$ , $8$ and $16$ regions, so you might conjecture that each new point doubles the count. With $6$ points, you get $31$ regions, not $32$ . The earlier cases did not prove the conjecture, and this one case disproves it.

Deductive reasoning goes from statements you accept as true to a new statement by using rules of logic. The starting statements can be given facts, definitions, postulates (statements accepted without proof) or theorems (statements already proved). When every starting statement is true and every step follows a rule of logic, the conclusion must be true. You do not sample cases, so the result holds in every case the starting statements cover.

Inductive and deductive reasoning compared

Question	Inductive reasoning	Deductive reasoning
What does it start from?	Specific examples or observations	Given facts, definitions, postulates or theorems
What does it give you?	A conjecture	A conclusion reached by rules of logic
Is the result proved?	No, however many examples agree	Yes, when every starting statement is true and every step follows a rule of logic
What about counterexamples?	One counterexample disproves the conjecture	None can exist when every starting statement is true and every step follows a rule of logic

Worked example

Classify the reasoning

Classify each argument as inductive or deductive. Argument A: Lena measures the angles of five triangles. Each time, the measures add to about $180^\circ$ , so she says that the angle measures of every triangle add to $180^\circ$ . Argument B: You are given that $\angle P$ is a right angle. By the definition of a right angle, if an angle is a right angle, then its measure is $90^\circ$ . So $m\angle P = 90^\circ$ .

Look at how Argument A reaches its claim. It starts with five measured triangles and extends the pattern to every triangle. Going from specific examples to a general conjecture is inductive reasoning.
Lena's claim is in fact true: it is the Triangle Angle Sum Theorem. Her five examples still do not prove it, because any number of confirming examples never proves a conjecture. So you classify an argument by how it reaches its claim, not by whether the claim is true.
Look at how Argument B reaches its claim. It starts with a given fact and the definition of a right angle, then applies logic without measuring anything. Going from accepted statements to a conclusion by rules of logic is deductive reasoning.

Answer

Argument A uses inductive reasoning, and Argument B uses deductive reasoning.

Argument B used a rule of logic that you will use often in proofs. Think of a conditional, an if-then statement, as a promise: whenever the hypothesis holds, the conclusion holds too. Write a conditional with hypothesis $p$ and conclusion $q$ as $p \to q$ , read "if $p$ , then $q$ ." When you know that the promise is true and that $p$ is true in your situation, you can separate $q$ from the conditional and state it on its own.

Key concept

The Law of Detachment

The Law of Detachment is a rule of deductive reasoning. If a conditional $p \to q$ is true and its hypothesis $p$ is true, then its conclusion $q$ must be true. You detach $q$ from the conditional and state it by itself. Both conditions are required. Unless you know both are true, this law gives you no conclusion.

p \to q \text{ is true and } p \text{ is true, so } q \text{ is true.}

Why must $q$ be true? A conditional is false only when its hypothesis is true and its conclusion is false, which gives the four cases in the table below. Since $p \to q$ is true, row 2 is out. Since $p$ is true, rows 3 and 4 are out. Only row 1 is left, and in that row $q$ is true.

When a conditional is true or false

Row	Hypothesis p	Conclusion q	Conditional: if p, then q
1	True	True	True
2	True	False	False
3	False	True	True
4	False	False	True

Worked example

Detach a conclusion from a theorem

Vertical angles are a pair of opposite angles formed when two lines intersect, and congruent angles are angles with equal measures. Given: $\angle 1$ and $\angle 2$ are vertical angles. True conditional (Vertical Angles Theorem): if two angles are vertical angles, then they are congruent. What can you conclude about $\angle 1$ and $\angle 2$ ?

Name the parts of the conditional. The hypothesis p is "two angles are vertical angles," and the conclusion q is "they are congruent."
Check the first condition. The conditional is true because it is the Vertical Angles Theorem, and it covers every pair of vertical angles. You are using it in its stated direction, from vertical angles to congruent angles. You are not using its converse, the statement made by swapping the hypothesis and conclusion.
Check the second condition. The hypothesis is true for this pair because the problem states that angle 1 and angle 2 are vertical angles.
Both conditions hold, so the Law of Detachment lets you state the conclusion for this pair.
$\angle 1 \cong \angle 2$

Answer

By the Law of Detachment, $\angle 1 \cong \angle 2$ , which means the two angles are congruent.

Worked example

Detach a conclusion, then solve

Given: $ABC$ is a triangle with $m\angle A = (2x + 10)^\circ$ , $m\angle B = (3x - 5)^\circ$ and $m\angle C = (x + 25)^\circ$ . True conditional (Triangle Angle Sum Theorem): if a figure is a triangle, then the measures of its interior angles add to $180^\circ$ . Find $x$ and each angle measure.

The conditional is true because it is the Triangle Angle Sum Theorem, used in its stated direction. Its hypothesis is true because the problem states that ABC is a triangle. The Law of Detachment gives the conclusion for this triangle.
$m\angle A + m\angle B + m\angle C = 180^\circ$
Substitute the given expressions, working in degrees, by the Substitution Property of Equality.
$(2x + 10) + (3x - 5) + (x + 25) = 180$
Group like terms by the Commutative and Associative Properties of Addition. Then combine them, using the Distributive Property for the x terms and arithmetic for the constants.
$(2x + 3x + x) + (10 - 5 + 25) = 180 \text{, so } 6x + 30 = 180$
Subtract 30 from both sides, by the Subtraction Property of Equality.
$6x + 30 - 30 = 180 - 30 \text{, so } 6x = 150$
Divide both sides by 6, which is not zero, by the Division Property of Equality.
$\frac{6x}{6} = \frac{150}{6} \text{, so } x = 25$
Substitute x = 25 into the expressions for angle A, angle B and angle C, in that order, by the Substitution Property of Equality. Check the result: each measure is between 0 and 180 degrees, and 60 + 70 + 50 = 180.
$2(25) + 10 = 60,\quad 3(25) - 5 = 70,\quad 25 + 25 = 50$

Answer

The value of $x$ is $25$ , and the angle measures are $m\angle A = 60^\circ$ , $m\angle B = 70^\circ$ and $m\angle C = 50^\circ$ .

Worked example

Decide whether the law applies

By definition, a square is a quadrilateral with four congruent sides and four right angles. So this conditional is true: if a quadrilateral is a square, then it has four right angles. Use the Law of Detachment with each given fact if you can. Fact A: Quadrilateral $ABCD$ is a square. Fact B: Quadrilateral $EFGH$ has four right angles. Fact C: Quadrilateral $JKLM$ is not a square.

Fact A states that the hypothesis is true for ABCD, and the conditional is true. Both conditions hold, so the Law of Detachment lets you state the conclusion.
$ABCD \text{ has four right angles}$
Fact B states the conclusion, not the hypothesis, so the Law of Detachment does not apply. Saying EFGH is a square would use the converse, which is false here: a rectangle with side lengths 3 and 5 has four right angles but is not a square.
Fact C states that the hypothesis is false, so the Law of Detachment does not apply. Rows 3 and 4 of the earlier table show why: when the hypothesis is false, the conditional is true whether its conclusion is true or false.
Test that with two quadrilaterals that are not squares. The rectangle with side lengths 3 and 5 has four right angles. A quadrilateral with angles of 60, 120, 60 and 120 degrees has no right angles. So you cannot tell whether JKLM has four right angles, and saying it does not would use the inverse, which is false here.

Answer

Only Fact A gives a conclusion by the Law of Detachment: $ABCD$ has four right angles. Facts B and C give no conclusion by this law.

Inductive reasoning finds conjectures, but no number of examples proves one, and one counterexample disproves it. Deductive reasoning uses rules of logic to reach conclusions that must be true when its starting statements are true. The Law of Detachment is one such rule: when $p \to q$ is true and $p$ is true, you can state $q$ . Next, you will use the Law of Syllogism and learn to determine whether a conclusion is valid.