Two-Column, Paragraph & Flow Proofs
A proof is a logical argument that starts from given information and ends at the statement you must prove, with every step justified along the way. Geometry gives you three formats for writing that argument: two-column, paragraph, and flow. The logic is identical in all three — only the layout changes.
The skill in this lesson is structural: know what every proof must contain, and recognize each format on sight. Once the pieces are clear, choosing a format is just choosing how to arrange them.
What every proof contains
Every proof has four parts: the given information, the statement to prove, a chain of statements connecting them, and a reason for each statement. The first line typically restates the given; the last line must match exactly what you were asked to prove.
The reasons are what make it a proof. A valid reason is a given, a definition, a postulate, a property, or a previously proven theorem — never “it looks true” or “it works in the diagram.” If a step has no reason, the argument has a hole in it.
The three formats
A two-column proof lists statements on the left and the reason for each statement directly across from it on the right. It is the most common school format because the structure forces you to justify every line. The table below shows the layout on a short vertical-angles argument.
A paragraph proof writes the same argument in sentences: “Since is the midpoint of , by the definition of midpoint...” The reasons are woven into the prose instead of sitting in their own column.
A flow proof puts each statement in a box with its reason underneath, and draws arrows showing which statements feed into which conclusions. It is the best format for seeing how two separate facts combine into one deduction.
How to read a two-column proof
Read across, then down. Each row says “this statement is true because of this reason,” and each new statement is allowed to use any statement above it. When you check someone else's proof — or your own — ask two questions at every row: does the reason actually justify this statement, and does this statement follow from what came before?
Worked examples
Example 1: a short two-column proof
Given: is the midpoint of . Prove: .
Answer: , justified by the definition of midpoint.
Example 2: identify the format
A proof is written as boxes connected by arrows, with a reason written under each box. Which format is it?
Answer: A flow proof
Example 3: spot the invalid reason
A student justifies the step “” with the reason “they look equal in the diagram.” Why is this not valid?
Answer: Appearance is not a valid reason; every statement needs a given, definition, postulate, property, or theorem.
Try one yourself
Common questions
Which proof format should I use?
Whichever the problem asks for. If you get to choose, two-column is usually easiest to organize because every statement is forced to sit next to its reason. The logic is the same in all three formats.
Does every single line need a reason?
Yes — including the first line, whose reason is “Given.” A statement without a reason is an assumption, and proofs are not allowed to assume.
Can I use information from the diagram?
Only what is marked or given. You may use the way points sit on a line or rays share a vertex, but never measurements or congruences that merely look true in the picture.
What goes in the last line of a proof?
Exactly the statement you were asked to prove, with a reason that justifies it. If your last line does not match the prove statement, the proof is not finished.
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