Learn it
A proof is a logical argument where every step has a reason: a given fact, a definition, a property or a theorem.
A two-column proof lists statements on the left and reasons on the right. Common reasons: Given; Definition of midpoint; Vertical angles are congruent; Reflexive property (AB = AB); Substitution; SAS, ASA, SSS; CPCTC.
Plan backwards: what do you need to show, and which facts lead there?
Worked examples
Example 1
Given: M is the midpoint of AB. Prove: AM = MB.
- Statement: M is the midpoint of AB. Reason: Given
- Statement: AM = MB. Reason: Definition of midpoint
Answer: Proved in two steps
Example 2
Given: AB = CD. Prove: AC = BD (A, B, C, D in order on a line).
- AB = CD (Given)
- AB + BC = CD + BC (Addition property)
- AC = AB + BC and BD = BC + CD (Segment addition)
- AC = BD (Substitution)
Answer: AC = BD
Practice
Try these on paper first, then check your work.
- Reason: ∠1 and ∠2 are vertical angles, so ∠1 = ∠2
- Reason: AB = AB
- If x + 3 = 7 then x = 4
- Reason for △s congruent with 3 equal sides
Show the answer key
- Vertical angles are congruent
- Reflexive property
- Subtraction property of equality
- SSS