Lesson 2.1.3

Conditional Statements (If-Then)

“If it rains, then the ground is wet.” Conditional statements are the language of logic — and the foundation of every geometric proof.

Introduction

Mathematics speaks in “if… then…” sentences. Understanding the structure of conditional statements lets you read theorems precisely and build airtight proofs.

Past Knowledge

Conjectures (2.1.1). Counterexamples (2.1.2).

Today's Goal

Identify the hypothesis and conclusion in conditional statements and determine truth values.

Future Success

Converse/inverse/contrapositive (2.1.4) and logical laws (2.2) all depend on this.

Key Concepts

Structure of a Conditional

A conditional statement has the form:

Hypothesis (): The “if” part — the condition.
Conclusion (): The “then” part — what follows.

Truth Value

A conditional is false only when the hypothesis is true and the conclusion is false. In all other cases, it is considered true.

Rewriting Common Statements

“All right angles are congruent” becomes: “If two angles are right angles, then they are congruent.”

Worked Examples

Basic

Identifying Parts

“If an angle measures 90°, then it is a right angle.” Identify the hypothesis and conclusion.

Hypothesis: An angle measures 90°.

Conclusion: It is a right angle.

Truth value: True.

Intermediate

Rewriting as If-Then

Rewrite: “Vertical angles are congruent.”

If two angles are vertical angles, then they are congruent.

Hypothesis: Two angles are vertical angles. Conclusion: They are congruent.

Advanced

False Conditional

“If a number is odd, then it is prime.” True or false?

Counterexample: 9 is odd but not prime ().

Answer: False — 9 is a counterexample.

Common Pitfalls

Confusing Hypothesis and Conclusion

The hypothesis always comes after “if” and the conclusion after “then.” Swapping them creates a completely different statement (the converse).

Real-Life Applications

Legal Contracts

Contracts are full of conditional logic: “If the payment is received by the 1st, then the service will continue.” Understanding hypothesis vs. conclusion helps you read and negotiate agreements clearly.