A two-column proof lists statements on the left and the reason for each on the right.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Start from the given
The first lines are what you were told. They need no other justification.
What counts as a reason
A definition, a postulate, or a theorem already proved. Never "it looks like it".
End at the goal
The final statement is exactly what was to be proved, and nothing more.
Claims on the left, reasons on the right
A two-column proof pairs every statement with its justification. The format exists to make the argument auditable: a reader can check each line against the reason given, without reconstructing your thinking.
What counts as a reason
A definition, a postulate, a previously proved theorem, or the given information. "It looks that way" and "obviously" are not reasons. Restricting yourself to legitimate reasons is what makes the exercise worth doing.
Plan backwards from the conclusion
Ask what would immediately give the conclusion, then what would give that. Working backwards from the goal usually finds the path faster than working forwards from the givens, and it is how experienced provers actually think.
Mark the diagram
Tick marks for equal sides, arcs for equal angles, and every deduction written on as you make it. A marked diagram often reveals the next step directly, and it prevents you from silently assuming something you have not established.
Step 2: Try It Yourself
Tap and try it out.
Triangle
Straight sides
3
Corners
3
A triangle has 3 straight sides.
Turning it does not change what it is.
Step 3: Watch an Example
One step at a time.
Watch Amara Prove Two Triangles Congruent
Given two sides and the included angle equal, Amara proves the triangles congruent.
- Step 1
She lists the two equal sides, with "given" as the reason for each.
Step 4: Your Turn
Practice makes it stick.
The Criterion
Problem 1 of 2
Two sides and the included angle. How many pairs are used? Give the count of matched pairs.
The Reason
Problem 2 of 2
Is "the diagram looks that way" a valid reason? 1 for yes, 0 for no.
Statements and Reasons
1 of 8
Order the lines of a two-column proof.
- 1Derive intermediate facts with reasons
- 2Name the criterion being applied
- 3State the conclusion
- 4State the given information
2 of 8
How many columns does a two-column proof have?
3 of 8
Is a definition a valid reason? 1 for yes, 0 for no.
4 of 8
Is an unproved conjecture a valid reason? 1 for yes, 0 for no.
5 of 8
Select every acceptable reason in a proof.
6 of 8
Does the reflexive property let a shared side equal itself? 1 for yes, 0 for no.
7 of 8
A proof concludes something not asked for. Is it complete? 1 for yes, 0 for no.
8 of 8
How many pairs must match for SSS?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Is "given" a valid reason for a first line? 1 for yes, 0 for no.
What You Learned
- A two-column proof pairs every statement with a reason.
- The given information opens the proof and needs no justification.
- Only definitions, postulates, and proved theorems count as reasons.