Two-Column Proofs
A free Geometry lesson from the “Coordinate Geometry and Proof” unit, with a worked example and practice problems including step-by-step solutions.
A proof starts with the givens and reaches the claim through definitions, established theorems, and justified algebra. A two-column format pairs every statement with its reason. Do not cite the theorem being proved as a reason. A diagram suggests relationships, but an unmarked visual resemblance is not a given.
What you'll learn
- Read a two-column proof, separating statements from reasons
- Identify common reasons (given, definition, theorem, postulate, properties)
- Order the steps of a proof so each statement follows from previous statements and reasons
Worked example
Problem. Distinct lines AB and CD intersect at O, with A and B on opposite rays and C and D on opposite rays. Prove angle AOC = angle BOD without assuming the vertical-angle theorem.
- Angle AOC + angle COB = 180 degrees because OA and OB are opposite rays (linear pair).
- Angle COB + angle BOD = 180 degrees because OC and OD are opposite rays (linear pair).
- The two sums are equal by transitivity of equality.
- Subtract angle COB from each sum. The result is angle AOC = angle BOD.
Answer: The opposite angles are equal because both are supplementary to angle COB.
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 a two-column proof, the left column holds:
Choices: Statements · Reasons
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- Statements describe what is known or being shown.
Answer: Statements
2. The right column holds:
Choices: Statements · Reasons
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- Each statement must be justified by a reason.
Answer: Reasons
3. The reason for the very first line of a proof is usually:
Choices: Given · Reflexive property
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- The starting facts come from the problem itself.
Answer: Given
4. The reason 'a = a' is the:
Choices: Reflexive property · Symmetric property · Transitive property
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- Anything equals itself.
Answer: Reflexive property
5. The reason 'if a = b then b = a' is the:
Choices: Reflexive property · Symmetric property · Transitive property
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- Equality goes both ways.
Answer: Symmetric property
Explain and apply
Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.
Produce a proof from givens
Draw two intersecting lines AB and CD with intersection O. Write a two-column proof that angle AOD = angle BOC using linear pairs and equality properties. Your reasons may not include the vertical-angle theorem. Then identify the circular step in: “The angles are vertical; vertical angles are equal; therefore vertical angles are equal.”
Review rubric and solution
- The diagram labels opposite rays correctly.
- Two supplementary sums share the same adjacent angle and are each justified by a linear pair.
- The conclusion follows by equating sums and subtracting the common angle.
- The circular argument is identified as using the desired theorem as a premise.
- AOD + DOB = 180 degrees, since OA and OB are opposite rays.
- DOB + BOC = 180 degrees, since OD and OC are opposite rays.
- Equate the sums and subtract DOB to obtain AOD = BOC.
- The quoted argument assumes the vertical-angle theorem in its second clause; it does not prove that theorem.
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