Topic 1 of 7
Logic and two-column proof
Step 12 · Geometry topics
Write the converse, inverse and contrapositive of a conditional statement, use counterexamples, name the reasons allowed in a proof, and write two-column proofs.
Statements and converses
A conditional statement has the form "if p, then q". p is the hypothesis and q is the conclusion. Rearranging it gives related statements, which may or may not be true.
| Statement | Form | Example | True? |
|---|---|---|---|
| conditional | if p, then q | If a shape is a square, it has 4 sides. | yes |
| converse | if q, then p | If a shape has 4 sides, it is a square. | no |
| inverse | if not p, then not q | If a shape is not a square, it does not have 4 sides. | no |
| contrapositive | if not q, then not p | If a shape does not have 4 sides, it is not a square. | yes |
Example: A counterexample
Is it true that if x² = 25, then x = 5?
- x = −5 also gives x² = 25.
AnswerNo: x = −5 is a counterexample
When a statement and its converse are both true, they combine into "p if and only if q", a biconditional. Good definitions are biconditionals: an angle is right if and only if it measures 90°.
Reasons you can use
Every step of a proof needs a reason: something given, a definition, a property of equality, a postulate (accepted without proof) or a theorem already proved.
| Reason | What it says |
|---|---|
| Given | stated in the problem |
| Definition of midpoint | a midpoint splits a segment into two equal parts |
| Reflexive property | AB = AB |
| Substitution | equal things can replace each other |
| Segment addition | if B is on AC, then AB + BC = AC |
| Vertical angles theorem | vertical angles are equal |
| Linear pair | two angles forming a straight line add to 180° |
Example: Naming the reason
A proof says ∠1 = ∠3 because ∠1 and ∠3 are vertical angles. What is the reason?
- The step uses the fact that vertical angles are equal.
AnswerThe vertical angles theorem
Writing a two-column proof
A two-column proof lists statements on the left and the reason for each on the right, starting from what is given and ending with what you want to prove.
Example: A segment proof
Given: M is the midpoint of AB, and AM = 7. Prove: AB = 14.
- 1. M is the midpoint of AB. Reason: given.
- 2. AM = MB. Reason: definition of midpoint.
- 3. MB = 7. Reason: substitution, since AM = 7.
- 4. AM + MB = AB. Reason: segment addition.
- 5. 7 + 7 is AB, so AB = 14. Reason: substitution.
AnswerAB = 14
Example: An angle proof
Given: ∠1 and ∠2 form a linear pair, and ∠1 = 65°. Prove: ∠2 = 115°.
- 1. ∠1 and ∠2 form a linear pair. Reason: given.
- 2. ∠1 + ∠2 = 180°. Reason: linear pair.
- 3. 65° + ∠2 = 180°. Reason: substitution.
- 4. ∠2 = 115°. Reason: subtraction property of equality.
Answer∠2 = 115°
Try it yourself
Work each one on paper first, then open the answer to check.
Write the converse of "If it is raining, then the ground is wet."
Show answer
If the ground is wet, then it is raining.
Is that converse always true?
Show answer
No: a sprinkler can wet the ground.
Write the contrapositive of "If x = 3, then 2x = 6."
Show answer
If 2x ≠ 6, then x ≠ 3.
Give a counterexample to "Every prime number is odd."
Show answer
2
Which reason justifies AB = AB?
Show answer
The reflexive property
Which reason justifies "if AB = CD and CD = EF, then AB = EF"?
Show answer
The transitive property (or substitution)