The Law of Syllogism and Valid Conclusions | Geometry | LevelKite

Good reasoning lets you reach facts that nobody told you directly. In this lesson, you will link if-then statements into a chain, and you will learn to decide whether a claimed conclusion really follows from what you are given.

An if-then statement is called a conditional: "if $p$ , then $q$ ," written $p \to q$ . The statement $p$ is its hypothesis, and $q$ is its conclusion. A conditional is false only when its hypothesis is true and its conclusion is false. In every other case, it is true. You already used a true conditional and its true hypothesis to get a new fact with the Law of Detachment, and now you will build on that.

Conditionals can connect like links in a chain. Look at two true statements about a number $x$ . If $x = 3$ , then $x + 2 = 5$ , by the Addition Property of Equality. If $x + 2 = 5$ , then $2x + 4 = 10$ , by the Multiplication Property of Equality.

The conclusion of the first statement is the hypothesis of the second, so truth passes through it. You can drop that middle statement and say directly: if $x = 3$ , then $2x + 4 = 10$ . The new statement does not claim that $x$ is $3$ . It says only what must be true when $x$ is $3$ .

Key concept

A valid conclusion is forced by the givens

A conclusion is valid when it must be true in every case where all the given statements are true. The Law of Syllogism is one law that forces a conclusion: if $p \to q$ is true and $q \to r$ is true, then $p \to r$ is true. The law applies only when both conditionals are true and the conclusion of the first matches the hypothesis of the second.

(p \to q) \text{ and } (q \to r) \;\Rightarrow\; (p \to r)

Here is why the law works. You are given that $p \to q$ and $q \to r$ are true, and you want to show that $p \to r$ is true. A conditional is false only when its hypothesis is true and its conclusion is false, so $p \to r$ could fail only in a case where $p$ is true and $r$ is false. Take any case where $p$ is true. Since $p \to q$ is true, the Law of Detachment gives $q$ . Since $q \to r$ is true, the Law of Detachment gives $r$ . So $r$ cannot be false in that case. No failing case exists, which means $p \to r$ is true.

Worked example

Link two conditionals

Given as true: (1) If two angles are vertical angles, then they are congruent. (2) If two angles are congruent, then they have equal measures. To find: the conditional that follows by the Law of Syllogism.

The law needs both statements to be true, and they are. Statement (1) is the Vertical Angles Theorem; vertical angles are the opposite angles formed when two lines intersect. Statement (2) comes from the definition of congruent angles, which are angles with equal measures.
The law also needs the conclusion of (1) to match the hypothesis of (2). Let p be "two angles are vertical angles," q be "they are congruent," and r be "they have equal measures." The statement q appears in both places, so it is the link.
$p \to q, \quad q \to r$
Apply the Law of Syllogism: keep the hypothesis of (1) and the conclusion of (2), and drop the link.
$p \to r$

Answer

By the Law of Syllogism, if two angles are vertical angles, then they have equal measures.

Worked example

Build a longer chain

Given as true: (1) If a quadrilateral is a parallelogram, then its opposite sides are congruent. (2) If a quadrilateral is a square, then it is a rectangle. (3) If a quadrilateral is a rectangle, then it is a parallelogram. To find: the conditional that links all three.

Put the statements in chain order, because each conclusion must match the next hypothesis. "Square" appears only in a hypothesis, so start with (2). Its conclusion matches the hypothesis of (3), and the conclusion of (3) matches the hypothesis of (1). Let p, q, r and s stand for "it is a square," "it is a rectangle," "it is a parallelogram" and "its opposite sides are congruent."
$p \to q, \quad q \to r, \quad r \to s$
Apply the Law of Syllogism to (2) and (3), because the law links two statements at a time. This gives a new true conditional: if a quadrilateral is a square, then it is a parallelogram.
$p \to q \text{ and } q \to r \;\Rightarrow\; p \to r$
Apply the law again to the new conditional and (1), because the conclusion r of the new conditional matches the hypothesis of (1).
$p \to r \text{ and } r \to s \;\Rightarrow\; p \to s$

Answer

By two uses of the Law of Syllogism, if a quadrilateral is a square, then its opposite sides are congruent.

Now for the second skill: deciding whether a conclusion is valid. Here conclusion means the statement someone claims follows from the givens, which is different from the then-part of a conditional. Until that claim is justified, it is a conjecture: a statement you think is true but have not yet justified.

Validity is about the reasoning. A conclusion that is not valid may still happen to be true in some cases; the givens just do not force it. A valid conclusion is guaranteed to be true only when the givens are true. Count only what is stated as given, because a fact that a drawing merely suggests is not a given.

To judge conclusions, you need three statements built from any conditional $p \to q$ . Here "not $p$ " means the statement that $p$ is false.

The converse swaps the hypothesis and the conclusion.
The inverse changes both to their "not" forms.
The contrapositive does both: it swaps them and changes both to their "not" forms.

A statement's truth value is whether it is true or false. A counterexample to a conditional is a specific case where its hypothesis is true and its conclusion is false. The table's example uses an acute angle, which is an angle with a measure between $0^\circ$ and $90^\circ$ . The symbol $m\angle A$ means the measure of $\angle A$ .

Four related statements and the one case in which each is false

Statement	Form	False only when	Example	Truth value of the example
Conditional	$p \to q$	$p$ true and $q$ false	If $m\angle A = 30^\circ$ , then $\angle A$ is acute.	True, by the definition of an acute angle
Contrapositive	not $q \to$ not $p$	$p$ true and $q$ false	If $\angle A$ is not acute, then $m\angle A \ne 30^\circ$ .	True: it fails only where the conditional fails, which never happens
Converse	$q \to p$	$q$ true and $p$ false	If $\angle A$ is acute, then $m\angle A = 30^\circ$ .	False: $m\angle A = 50^\circ$ is a counterexample
Inverse	not $p \to$ not $q$	$q$ true and $p$ false	If $m\angle A \ne 30^\circ$ , then $\angle A$ is not acute.	False: $m\angle A = 50^\circ$ is a counterexample

The third column shows why the truth values pair up. A conditional and its contrapositive fail in exactly the same case, so they always share a truth value: both are true or both are false. The converse and the inverse fail in the same case as each other, so they also share a truth value. That case differs from the case where the original fails, so their truth value may differ from the original's. In this table, it does differ. For another conditional, the converse and the inverse could both be true, but you would need a separate reason to say so. A true conditional does not give you its converse or inverse.

To decide whether a conclusion is valid, use this check.

List what is given: each conditional and each stated fact.
Try to reach the conclusion with a law: the Law of Detachment, the Law of Syllogism, or a contrapositive. You may use a contrapositive in place of its conditional because the two always share a truth value. If a law reaches the conclusion, it is valid.
If no law reaches it, look for a counterexample to the conclusion: one case where every given is true and the conclusion is false. One counterexample is enough to show that the conclusion is not valid.

One counterexample disproves a conjecture, but any number of confirming examples never proves it. So a list of examples that agree cannot replace step 2. If you cannot complete step 2 or step 3 yet, the conclusion remains a conjecture.

Common patterns, judged using only the givens in each row

Given as true	Claimed conclusion	What the reasoning uses	Valid?
$p \to q$ and $p$	$q$	Law of Detachment	Valid
$p \to q$ and $q \to r$	$p \to r$	Law of Syllogism	Valid
$p \to q$ and not $q$	not $p$	Contrapositive, then Law of Detachment	Valid
$p \to q$ and $q$	$p$	Converse, which may be false	Not valid
$p \to q$ and not $p$	not $q$	Inverse, which may be false	Not valid

Worked example

Judge four conclusions

Given as true: If a quadrilateral is a square, then it has four right angles. To decide: is each conclusion valid? (a) $ABCD$ is a square, so it has four right angles. (b) $EFGH$ has four right angles, so it is a square. (c) $JKLM$ is not a square, so it does not have four right angles. (d) $WXYZ$ does not have four right angles, so it is not a square.

Name the parts of the given conditional so you can compare each conclusion with it. Let p be "the quadrilateral is a square" and q be "it has four right angles."
$p \to q$
Conclusion (a) starts from the stated fact p, which matches the hypothesis. The Law of Detachment gives q, so (a) is valid.
$p \to q \text{ and } p \;\Rightarrow\; q$
Conclusion (b) goes from q to p, so it relies on the converse, which may be false. Suppose EFGH is a rectangle with side lengths 3 and 5. It has four right angles, but its sides are not all congruent, so it is not a square. The given conditional still holds, because its hypothesis is false here. Every given is true and the conclusion is false, so this counterexample shows (b) is not valid.
$q \to p \;\text{(converse)}$
Conclusion (c) goes from not p to not q, so it relies on the inverse, which may also be false. Suppose JKLM is that same rectangle. It is not a square, yet it has four right angles. Every given is true and the conclusion is false, so (c) is not valid.
$\text{not } p \to \text{not } q \;\text{(inverse)}$
Conclusion (d) starts from the stated fact not q. The contrapositive, if not q then not p, always shares the truth value of the given conditional, so it is true. Its hypothesis matches the stated fact, so the Law of Detachment gives not p, and (d) is valid.
$\text{not } q \to \text{not } p \text{ and not } q \;\Rightarrow\; \text{not } p$

Answer

Conclusions (a) and (d) are valid, and conclusions (b) and (c) are not valid.

Worked example

Chain, detach, then solve

Given as true: (1) If two sides of a triangle are congruent, then the angles opposite those sides are congruent (Isosceles Triangle Theorem). (2) If two angles are congruent, then they have equal measures (definition of congruent angles). Stated facts: in $\triangle ABC$ , $\overline{AB} \cong \overline{AC}$ , $m\angle B = (3x)^\circ$ and $m\angle C = (x + 50)^\circ$ . To decide: is the conclusion $x = 25$ valid?

Link the two conditionals. Statement (2) covers any two angles, so its hypothesis matches the conclusion of (1). The Law of Syllogism gives: if two sides of a triangle are congruent, then the angles opposite those sides have equal measures. This uses the theorem in its forward direction, from sides to angles, not its converse.
The stated fact that sides AB and AC are congruent matches the hypothesis. Side AB is opposite angle C, and side AC is opposite angle B. The Law of Detachment gives equal measures for those angles.
$m\angle B = m\angle C$
Replace each measure with its given expression, by the Substitution Property of Equality. Both measures are in degrees, so the numbers are equal.
$3x = x + 50$
Subtract x from both sides, by the Subtraction Property of Equality.
$2x = 50$
Divide both sides by 2, by the Division Property of Equality.
$x = 25$
Check the value against the geometry. Both angles measure 75 degrees, which is between 0 and 180 degrees. Their sum is 150 degrees, so the Triangle Angle Sum Theorem leaves a positive 30 degrees for angle A.
$3(25) = 75, \quad 25 + 50 = 75$

Answer

The conclusion $x = 25$ is valid, because the Law of Syllogism, the Law of Detachment and the properties of equality force it from the givens.

Here is the idea to keep: a conclusion is valid only when the givens force it. The Law of Syllogism forces one when two true conditionals link, with the conclusion of the first matching the hypothesis of the second. A contrapositive always shares its conditional's truth value, but a converse or inverse may differ. Use a law to show that a conclusion is valid, and use one counterexample to show that it is not, because confirming examples never prove it.