Algebraic and Geometric Proof
A proof is a chain of statements, each justified by a property, definition, or given fact. The properties of equality that justify every step of solving an equation are the same ones used in geometric proofs.
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A proof is a chain of statements, each one justified by a property, definition, or previously established fact. Algebraic proofs use the properties of equality to justify each step of solving an equation; geometric proofs use the same properties alongside geometric definitions and postulates.
Concept
The properties of equality
| Property | Statement |
|---|---|
| Addition | If a = b, then a + c = b + c |
| Subtraction | If a = b, then a − c = b − c |
| Multiplication | If a = b, then ac = bc |
| Division | If a = b and c ≠ 0, then a/c = b/c |
| Distributive | a(b + c) = ab + ac |
| Reflexive | a = a |
| Symmetric | If a = b, then b = a |
| Transitive | If a = b and b = c, then a = c |
| Substitution | If a = b, then a can replace b in any expression |
A proof is simply a sequence of statements where every single line is backed by one of these properties, a definition, or a given fact — nothing is allowed to appear without justification.
Example
An algebraic proof
Given: . Prove that
.
| Statement | Reason |
|---|---|
| 3x + 7 = 22 | Given |
| 3x = 15 | Subtraction property of equality |
| x = 5 | Division property of equality |
Every algebra problem you have ever solved is already a proof — you simply were not writing the justification column beside each step.
Example
A proof using the distributive property
Given: . Prove that
.
4(x − 2) = 20 — Given
4x − 8 = 20 — Distributive property
4x = 28 — Addition property of equality
x = 7 — Division property of equality
Example
A geometric proof
Given: and
, where
and
. Prove that
.
| Statement | Reason |
|---|---|
| ∠1 ≅ ∠2, ∠2 ≅ ∠3 | Given |
| ∠1 ≅ ∠3 | Transitive property |
| m∠1 = m∠3 | Definition of congruent angles |
| 5x + 10 = 40 | Substitution |
| 5x = 30 | Subtraction property of equality |
| x = 6 | Division property of equality |
The first two lines use only geometric facts and definitions; from that point on, the proof is identical in structure to a purely algebraic one — the same properties of equality carry the argument through to the end.
Summary
- A proof is a sequence of statements, each justified by a property, definition, or given fact.
- Every step of solving an equation is already a small proof, using the addition, subtraction, multiplication, and division properties of equality.
- The transitive and substitution properties are especially useful for connecting separate given facts into a single chain.
- A geometric proof uses the same algebraic properties, plus geometric definitions and postulates, to justify each step.