Logic, Conditionals, and Biconditionals
A free Geometry lesson from the “Geometry Foundations” unit, with a worked example and practice problems including step-by-step solutions.
A conditional statement has the form 'if p, then q' where p is the hypothesis and q is the conclusion. Its converse is 'if q, then p'; its inverse is 'if not p, then not q'; its contrapositive is 'if not q, then not p'. A conditional and its contrapositive are always logically equivalent. A biconditional 'p if and only if q' is true exactly when both the conditional and its converse are true.
What you'll learn
- Identify the hypothesis and conclusion of a conditional statement
- Write the converse, inverse, and contrapositive of a conditional
- Recognize biconditionals and when they apply
Worked example
Problem. Write the converse of 'If a figure is a square, then it has four sides.'
- Swap the hypothesis ('a figure is a square') and the conclusion ('it has four sides').
- Converse: 'If a figure has four sides, then it is a square.'
- Note: this converse is FALSE — a rectangle has four sides too.
- Connect the result back to Logic, Conditionals, and Biconditionals so the geometric relationship is explicit.
Answer: If a figure has four sides, then it is a square.
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. In 'If it rains, then the ground is wet,' what is the hypothesis?
Choices: It rains · The ground is wet
Show solution
- The 'if' clause is the hypothesis.
Answer: It rains
2. In the same statement, what is the conclusion?
Choices: It rains · The ground is wet
Show solution
- The 'then' clause is the conclusion.
Answer: The ground is wet
3. What is the converse of 'If p, then q'?
Choices: If q, then p · If not p, then not q · If not q, then not p
Show solution
- Converse swaps hypothesis and conclusion.
Answer: If q, then p
4. What is the inverse of 'If p, then q'?
Choices: If q, then p · If not p, then not q · If not q, then not p
Show solution
- Inverse negates both parts but keeps the order.
Answer: If not p, then not q
5. What is the contrapositive of 'If p, then q'?
Choices: If q, then p · If not p, then not q · If not q, then not p
Show solution
- Contrapositive negates AND swaps.
Answer: If not q, then not p
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