Inductive reasoning uses patterns in specific cases to form a general conclusion called a conjecture. It is the basis of mathematical discovery — but inductive reasoning can lead to false conclusions.
Inductive reasoning: Observing a pattern and drawing a general conclusion.
Conjecture: A conclusion based on observed patterns. Not yet proven — it may be false.
Counterexample: A single specific case that proves a conjecture false. One counterexample is enough to disprove any conjecture.
Pattern: 1, 4, 9, 16, 25, … (perfect squares)
Conjecture: "The difference between consecutive perfect squares is always odd."
4−1=3 ✓, 9−4=5 ✓, 16−9=7 ✓, 25−16=9 ✓ → Pattern holds so far.
Is the conjecture true? In general, $(n+1)^2 - n^2 = 2n+1$, which is always odd. This conjecture happens to be true — but inductive reasoning alone doesn't prove it; we need deductive reasoning.
A false conjecture example: "The expression $n^2 - n + 41$ is always prime." It works for $n=1$ through $n=40$, but at $n=41$: $41^2 - 41 + 41 = 41^2 = 1681$, which is not prime. Counterexample found.
A student conjectures: "The product of any two odd numbers is odd." Find three examples that support this, then determine if it's actually true or find a counterexample.
A conditional statement (also called an if-then statement) has the form: If p, then q, written $p \to q$. The part after "if" is the hypothesis (p); the part after "then" is the conclusion (q).
Key fact: A conditional and its contrapositive always have the same truth value. The converse and inverse always have the same truth value (but may differ from the original).
Original: "If it is raining, then the ground is wet." (True)
| Statement | Form | Truth |
|---|---|---|
| If raining, then wet | $p \to q$ | True |
| If wet, then raining (Converse) | $q \to p$ | False* |
| If not raining, then not wet (Inverse) | $\neg p \to \neg q$ | False* |
| If not wet, then not raining (Contrapositive) | $\neg q \to \neg p$ | True |
*A sprinkler could wet the ground without rain — counterexample disproves the converse.
A biconditional statement combines a conditional and its converse: "p if and only if q" (written $p \leftrightarrow q$). It is true only when both $p$ and $q$ have the same truth value.
A biconditional is valid (true) when both the conditional and its converse are true. Definitions in geometry are always biconditionals.
"An angle is a right angle if and only if its measure is 90°."
This is valid because: (1) If an angle is right, its measure is 90° ✓, and (2) If its measure is 90°, it is a right angle ✓.
Invalid biconditional attempt: "A figure is a square if and only if it has four sides." The converse fails — a rectangle has four sides but is not necessarily a square. So this biconditional is false.
Conditional p→q illustrated: when p is true, q must also be true
Figure 2.1 — Venn Diagram: Conditional Statement p → q
Deductive reasoning uses accepted facts, definitions, postulates, and theorems to reach conclusions that must be true. It is the foundation of mathematical proof.
Law of Detachment: If $p \to q$ is true and $p$ is true, then $q$ must be true.
Example: "If it rains, the game is cancelled." It is raining. Therefore, the game is cancelled.
Law of Syllogism: If $p \to q$ and $q \to r$ are both true, then $p \to r$ is true.
Example: "If I study, I pass. If I pass, I graduate." Therefore: "If I study, I graduate."
Statement 1: If a number is divisible by 6, then it is divisible by 3.
Statement 2: If a number is divisible by 3, then the sum of its digits is divisible by 3.
Conclusion (Law of Syllogism): If a number is divisible by 6, then the sum of its digits is divisible by 3.
Given: (1) If $\angle A$ and $\angle B$ are supplementary, then $m\angle A + m\angle B = 180°$. (2) $\angle A$ and $\angle B$ are supplementary. What can you conclude? What law did you use?
Proofs in geometry use properties of equality (for measures) and congruence (for geometric figures). These are the tools that justify each step.
A two-column proof organizes statements and reasons in two columns. Each statement must be justified by a definition, property, postulate, or theorem.
Proof Strategy: Start by writing "Given" as your first statement. Identify what you need to prove. Work through the chain of logic — each new statement should follow from what you've established, ending with the "Prove" statement as your last line.
Given: $2(x + 3) = 14$
Prove: $x = 4$
| Statements | Reasons |
|---|---|
| 1. $2(x + 3) = 14$ | Given |
| 2. $2x + 6 = 14$ | Distributive Property |
| 3. $2x = 8$ | Subtraction Property of Equality |
| 4. $x = 4$ | Division Property of Equality |
Given: $\overline{AC}$ with $B$ between $A$ and $C$; $AB = CD$; $B$, $C$, $D$ are collinear with $C$ between $B$ and $D$
Prove: $AC = BD$
| Statements | Reasons |
|---|---|
| 1. $B$ is between $A$ and $C$; $C$ is between $B$ and $D$ | Given |
| 2. $AC = AB + BC$ | Segment Addition Postulate |
| 3. $BD = BC + CD$ | Segment Addition Postulate |
| 4. $AB = CD$ | Given |
| 5. $AC = BC + CD$ | Substitution (steps 2 and 4) |
| 6. $AC = BD$ | Transitive Property of Equality (steps 3 and 5) |
Segment Addition Postulate: B between A and C means AB + BC = AC
Figure 2.2 — Segment Addition Postulate Visualization
Given: $m\angle 1 + m\angle 2 = 90°$ and $m\angle 2 = 35°$. Prove: $m\angle 1 = 55°$ using a two-column proof.
| Statements | Reasons |
|---|---|
| 1. m∠1 + m∠2 = 90° | Given |
| 2. m∠2 = 35° | Given |
| 3. m∠1 + 35° = 90° | Substitution (steps 1, 2) |
| 4. m∠1 = 55° | Subtraction Property of Equality |
Complementary angles: when two angles sum to 90°
Figure 2.3 — Complementary Angles and Angle Addition
Find a counterexample to disprove: "If $n$ is a whole number, then $n^2 > n$."
Write the converse, inverse, and contrapositive of: "If a polygon is a square, then it has four sides." Determine the truth value of each.
Use the Law of Syllogism to write a new conditional: (1) "If it is Friday, then we have pizza." (2) "If we have pizza, then everyone is happy."
Is the following a valid biconditional? "A triangle is equilateral if and only if all three angles are 60°." Explain.
Complete the proof. Given: $3x - 7 = 20$. Prove: $x = 9$. Fill in the missing reasons.
Given that $\angle 1 \cong \angle 3$ and $\angle 3 \cong \angle 5$, what can you conclude? What property justifies your conclusion?
Write a two-column proof. Given: $\angle ABC$ is a right angle. $\angle ABD + \angle DBC = \angle ABC$. $\angle ABD = 40°$. Prove: $\angle DBC = 50°$.
Look at the pattern: 2, 5, 10, 17, 26, … Make a conjecture about the next two terms and describe the pattern.
Drawing a general conclusion from specific examples. Results in a conjecture. Not guaranteed to be true — one counterexample disproves it.
Using accepted facts (postulates, theorems) to prove new conclusions. The basis of geometric proof. If premises are true, the conclusion must be true.
"If $p$, then $q$" — written $p \Rightarrow q$. The hypothesis is $p$; conclusion is $q$.
"If $q$, then $p$" — $q \Rightarrow p$. Not necessarily true even when the original is true.
"If not $p$, then not $q$" — $\neg p \Rightarrow \neg q$. Same truth value as the converse.
"If not $q$, then not $p$" — $\neg q \Rightarrow \neg p$. Always has the same truth value as the original conditional.