A proof starts from the given information and reaches the required conclusion, justifying every step.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Two columns
Statements go on the left and reasons on the right. Every statement needs a reason, and "it looks true" is not one.
What counts as a reason
Given information, a definition, or a previously established theorem. Nothing else.
The diagram is not a reason
Two segments looking equal proves nothing. Only stated or derived facts may be used.
Work backwards to plan
Ask what would give the conclusion, then what would give that. Planning backwards and writing forwards is the usual method.
A common shape of proof
Many proofs establish two triangles congruent, then conclude that corresponding parts are equal.
Every claim carries its reason
A two-column proof pairs each statement with its justification. The format makes the argument auditable: a reader can check each line without reconstructing your reasoning.
Plan backwards from the conclusion
Ask what would immediately give the result, then what would give that. Working backwards from the goal usually finds the path faster than working forwards from the givens.
Mark the diagram as you go
Tick marks for equal sides, arcs for equal angles, and every deduction written on. A marked diagram often reveals the next step directly and prevents silent assumptions.
What is admissible as a reason
Definitions, postulates, proved theorems and the given information. "It looks that way" is not a reason, and restricting yourself to legitimate ones is what makes the exercise worthwhile.
Step 2: Try It Yourself
Tap and try it out.
The hypotenuse is not given. It comes from 6² + 8² = 100, whose square root is 10.
Step 3: Watch an Example
One step at a time.
Watch Ines Plan a Proof
Ines must prove two segments equal, given two triangles sharing a side and two pairs of equal sides.
- Step 1
She works backwards: equal corresponding parts would follow from congruent triangles.
Step 4: Your Turn
Practice makes it stick.
The Shared Side
Problem 1 of 2
A side shared by two triangles is equal to itself. How many pairs of equal sides does that supply?
The Criterion
Problem 2 of 2
Three pairs of equal sides. Which criterion? 1 SSS, 2 SAS, 3 ASA.
Justify Each Step
1 of 8
Is "the diagram looks that way" a valid reason? 1 yes, 0 no.
2 of 8
Is a previously proved theorem a valid reason? 1 yes, 0 no.
3 of 8
Two triangles with two equal sides and an equal included angle. Which criterion? 1 SSS, 2 SAS, 3 ASA.
4 of 8
How many statements in a two-column proof need a reason?
5 of 8
A triangle has angles 70° and 60°. Third angle, in degrees?
6 of 8
Once two triangles are congruent, are corresponding parts equal? 1 yes, 0 no.
7 of 8
Put the proof-writing process in order.
- 1Work backwards to find what would give that conclusion.
- 2Write the statements forwards from the given facts.
- 3Attach a reason to every statement.
- 4Read the given information and the conclusion wanted.
8 of 8
Two angles and the included side equal. Which criterion? 1 SSS, 2 SAS, 3 ASA.
Step 5: Quick Check
Show what you know.
Question 1 of 2
Three pairs of equal sides. Which criterion? 1 SSS, 2 SAS, 3 ASA.
Question 2 of 2
Why can a diagram not justify a step?
What You Learned
- A proof runs from given information to a conclusion, justifying every step.
- Reasons are given facts, definitions, or established theorems.
- Plan backwards from the conclusion, then write forwards.