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 two-column proof has statements in the left column and the reason justifying each statement in the right column. Each statement must follow from previous statements and an explicit reason (given, definition, postulate, theorem, or property like reflexive / symmetric / transitive). The proof ends when the goal statement is reached.
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. Prove that if angles 1 and 2 are vertical angles, then they are congruent (key reason in the last step).
- Statement 1: Angles 1 and 2 are vertical angles. Reason: Given.
- Statement 2: They are formed by two intersecting lines. Reason: Definition of vertical angles.
- Statement 3: Angle 1 is congruent to angle 2. Reason: Vertical Angles Theorem.
Answer: Vertical Angles Theorem
Practice problems
1. In a two-column proof, the left column holds:
Choices: Statements · Reasons
Show solution
- Warm-up: First identify exactly what the question is asking: In a two-column proof, the left column holds:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Statements describe what is known or being shown.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Statements
2. The right column holds:
Choices: Statements · Reasons
Show solution
- Warm-up: First identify exactly what the question is asking: The right column holds:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Each statement must be justified by a reason.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Reasons
3. The reason for the very first line of a proof is usually:
Choices: Given · Reflexive property
Show solution
- Warm-up: First identify exactly what the question is asking: The reason for the very first line of a proof is usually:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The starting facts come from the problem itself.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Given
4. The reason 'a = a' is the:
Choices: Reflexive property · Symmetric property · Transitive property
Show solution
- Core Practice: First identify exactly what the question is asking: The reason 'a = a' is the:
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Anything equals itself.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Reflexive property
5. The reason 'if a = b then b = a' is the:
Choices: Reflexive property · Symmetric property · Transitive property
Show solution
- Core Practice: First identify exactly what the question is asking: The reason 'if a = b then b = a' is the:
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Equality goes both ways.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Symmetric property
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