Same Math › Geometry · grades 9–10 › Two-column proofs
How do two-column proofs work?
A two-column proof is an argument written as a table. The left column lists statements, one step at a time, from the given facts to what you're proving. The right column gives the reason for each step: “Given,” a definition, a property like “Subtraction Property of Equality,” or a theorem. No step is allowed without a reason.
- Statements: what you know, one fact per line.
- Reasons: why each line is true.
- First line: the given. Last line: what you're proving.
Why schools teach it
It teaches how to back up every claim
Geometry students prove theorems about lines, angles and triangles (HSG.CO.C.9 and C.10). Many classes start with algebra proofs, like solving 3x + 5 = 20 with a reason for each step, before moving to diagrams.
Parents often freeze because the reasons have exact names. They come from a short list the class builds; a reason bank on the wall or in the notebook is normal and expected.
At the kitchen table
What to say, and what to skip
Say this
What's given? Start there.
How did you get from this line to the next?
That's the reason.Is that a definition, a property or a theorem?
Check the class reason bank. Which name fits?
Skip this
- Skipping “obvious” steps.Even BC = BC needs a reason: the Reflexive Property.
- Writing a reason that's really a statement.Reasons are rules, like “Definition of midpoint.”
- Starting from what you're trying to prove.Start from the given and work toward it.
See it in action
Proving: if 2(x − 3) = 10, then x = 8
2(x − 3) = 10
Given2x − 6 = 10
Distributive Property2x = 16
Addition Property of Equalityx = 8
Division Property of Equality
Algebra proofs use the same properties kids already use to solve equations, now with names.
Free practice sheet
Make a two-column proof sheet
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Parents: tell your child's teacher about this guide
Parents also ask
Where do the reason names come from?
Each class keeps a list: properties of equality, definitions (midpoint, bisector), postulates and theorems. Wording varies a little by textbook.
What's the Reflexive Property?
Anything equals itself: AB = AB. It's used when two triangles share a side.
What's the difference between = and ≅?
= compares numbers (lengths, measures). ≅ means congruent: same size and shape, used for segments, angles and figures.
Are there other kinds of proofs?
Yes: flowchart proofs and paragraph proofs say the same thing in a different layout.