A conditional statement “If p, then q” has a converse, inverse and contrapositive, and only the contrapositive always has the same truth value. Inductive reasoning finds patterns, deductive reasoning proves them. A two-column proof lists statements with reasons, as in the proof that vertical angles are congruent.
Key facts:
- A conditional “If p, then q” has a hypothesis (p) and a conclusion (q).
- The converse swaps p and q, the inverse negates both, and the contrapositive does both.
- A statement and its contrapositive always have the same truth value.
- One counterexample disproves a conditional; inductive patterns are not proofs.
- Vertical angles are congruent, proved with linear pairs and the Subtraction Property of Equality.
What is a conditional statement?
A conditional statement has the form “If p, then q.” The if part is the hypothesis and the then part is the conclusion. Example: “If a polygon is a square, then it is a rectangle.” A conditional is false when one case has a true hypothesis and a false conclusion. That case is a counterexample.
How do you write the converse, inverse and contrapositive?
| Statement | Form | Square example |
|---|---|---|
| Conditional | If p, then q | If a polygon is a square, then it is a rectangle. (true) |
| Converse | If q, then p | If a polygon is a rectangle, then it is a square. (false) |
| Inverse | If not p, then not q | If a polygon is not a square, then it is not a rectangle. (false) |
| Contrapositive | If not q, then not p | If a polygon is not a rectangle, then it is not a square. (true) |
A statement and its contrapositive always have the same truth value. When a conditional and its converse are both true, combine them into a biconditional using “if and only if.”
What is the difference between inductive and deductive reasoning?
Inductive reasoning forms a conjecture from patterns. The squares below suggest that the sum of the first n odd numbers is \(n^{2}\), but patterns can fail, and one counterexample disproves a conjecture. Deductive reasoning uses accepted facts (definitions, postulates and proven theorems) in logical steps, so the conclusion must be true.
How do you write a two-column proof?
Reflexive: \(a=a\). Symmetric: if \(a=b\), then \(b=a\). Transitive: if \(a=b\) and \(b=c\), then \(a=c\).
Addition, Subtraction, Multiplication and Division Properties: doing the same thing to both sides keeps an equation true. Substitution: replace a quantity with an equal one.
- Mark the figure, then write the Given and the Prove.
- Write each statement in the left column and its reason in the right column.
- Every reason is Given, a definition, a postulate, a property or a proven theorem. The last statement is the Prove.
Worked example. Given: lines ℓ and m intersect. Prove: \(\angle 1\) ≅ \(\angle 3\) (vertical angles are congruent).
| Statements | Reasons |
|---|---|
| 1. Lines ℓ and m intersect | Given |
| 2. \(\angle 1\) and \(\angle 2\) are a linear pair; \(\angle 2\) and \(\angle 3\) are a linear pair | Definition of linear pair |
| 3. \(\angle 1\) and \(\angle 2\) are supplementary; \(\angle 2\) and \(\angle 3\) are supplementary | Linear Pair Postulate |
| 4. \(m\angle 1+m\angle 2=180^{\circ}\); \(m\angle 2+m\angle 3=180^{\circ}\) | Definition of supplementary angles |
| 5. \(m\angle 1+m\angle 2=m\angle 2+m\angle 3\) | Transitive Property of Equality |
| 6. \(m\angle 1=m\angle 3\) | Subtraction Property of Equality |
| 7. \(\angle 1\) ≅ \(\angle 3\) | Definition of congruent angles |
A paragraph proof gives the same steps in sentences.
Logic and Two-Column Proofs: frequently asked questions
How do you find the converse of a conditional statement?
Swap the hypothesis and the conclusion. The converse of “If it is a square, then it is a rectangle” is “If it is a rectangle, then it is a square.” A true statement can have a false converse.
Why does the contrapositive matter?
The contrapositive always has the same truth value as the original statement. If a statement is true, its contrapositive is true, which makes it useful in proofs.
What is the difference between a conjecture and a theorem?
A conjecture is a guess based on patterns, found by inductive reasoning. A theorem is a statement that has been proven with deductive reasoning from definitions, postulates and earlier theorems.
How many counterexamples disprove a statement?
Just one. A single case where the hypothesis is true and the conclusion is false shows that a conditional statement is false, even if it works in many other cases.
What reasons can you use in a two-column proof?
Use Given information, definitions, postulates, properties of equality and theorems that were already proven. Never cite a statement you are trying to prove.
See every chapter on the Geometry math page.