DWG NO. 02.2 — Lesson 2 of 5

Conditional Statements, Converses, and Biconditionals

Unit 2: Reasoning and Proof · ~15–30 min

Objective

Rewrite statements in if-then form, identify the hypothesis and conclusion, and write the converse, inverse, contrapositive, and biconditional of a conditional.

Conditional statements

A conditional statement has the form “if p, then q,” written symbolically as p → q. The part after “if” is the hypothesis (p); the part after “then” is the conclusion (q). For example: “If two angles are vertical angles, then they are congruent” has hypothesis “two angles are vertical angles” and conclusion “they are congruent.”

Related conditionals

Every conditional has three related statements, built by switching or negating its parts:

Name	Form	Built by
Conditional	p → q	original statement
Converse	q → p	switching hypothesis and conclusion
Inverse	~p → ~q	negating both parts
Contrapositive	~q → ~p	switching and negating both parts

The contrapositive always has the same truth value as the original conditional — they're logically equivalent. The converse and inverse are also equivalent to each other, but not necessarily to the original.

Dashed diagonals mark logically equivalent pairs: conditional↔contrapositive, and converse↔inverse.

Biconditional statements

When a conditional and its converse are both true, you can combine them into one statement using “if and only if” (often shortened to “iff”), written p ↔ q. For example: “Two angles are supplementary if and only if their measures sum to 180°” — both directions hold, so the biconditional is valid.

Worked Example 1 · If-then form

ProblemRewrite in if-then form and identify the hypothesis and conclusion: “A triangle with three congruent sides is equilateral.”

1.Find the condition being described and the result that follows from it.

2.If-then form: “If a triangle has three congruent sides, then it is equilateral.”

Hypothesis: a triangle has three congruent sides • Conclusion: it is equilateral

Worked Example 2 · Converse, inverse, contrapositive

ProblemConditional: “If a number is divisible by 4, then it is divisible by 2.” This is true.

1.Converse: “If a number is divisible by 2, then it is divisible by 4.” False — 6 is divisible by 2 but not by 4.

2.Inverse: “If a number is not divisible by 4, then it is not divisible by 2.” False — 6 is not divisible by 4, but it is divisible by 2.

3.Contrapositive: “If a number is not divisible by 2, then it is not divisible by 4.” True — matches the original conditional's truth value, as it always must.

Conditional: true • Converse: false • Inverse: false • Contrapositive: true

Guided practice

1.Rewrite “Vertical angles are congruent” in if-then form.
Show answer
If two angles are vertical angles, then they are congruent.
2.Identify the hypothesis and conclusion: “If a polygon has five sides, then it is a pentagon.”
Show answer
Hypothesis: a polygon has five sides. Conclusion: it is a pentagon.
3.Write the converse of “If a figure is a square, then it is a rectangle,” and state whether the converse is true.
Show answer
Converse: “If a figure is a rectangle, then it is a square.” False — a 2-by-4 rectangle is a counterexample; it's a rectangle but not a square.
4.Write the contrapositive of “If it is raining, then the ground is wet.”
Show answer
If the ground is not wet, then it is not raining.
5.Given “If a shape is a triangle, then it has three sides” and its converse “If a shape has three sides, then it is a triangle,” is the biconditional “A shape is a triangle if and only if it has three sides” valid? Explain.
Show answer
Yes. Both the conditional and its converse are true, so they can be combined into a valid biconditional.

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