Properties of Equality & Congruence
The rules you've used since algebra — addition property, substitution, transitive — now become reasons in a geometry proof.
Introduction
Every step in a proof must be justified by a reason. This lesson catalogues the algebraic and geometric properties you'll cite most often.
Past Knowledge
Algebra properties. Deductive reasoning (2.2).
Today's Goal
Name and apply the properties of equality and congruence used in proofs.
Future Success
Every two-column proof (2.3.3) and beyond cites these properties as reasons.
Key Concepts
Properties of Equality
| Property | Statement |
|---|---|
| Reflexive | |
| Symmetric | If , then |
| Transitive | If and , then |
| Addition | If , then |
| Subtraction | If , then |
| Multiplication | If , then |
| Division | If and , then |
| Substitution | If , then can replace in any expression |
Properties of Congruence
Congruence () has its own reflexive, symmetric, and transitive properties — they mirror equality but apply to geometric figures, not numbers.
Worked Examples
Naming the Property
If and , then . Name the property.
Answer: Transitive Property of Equality.
Justifying a Step
If , then . What property was used?
Answer: Subtraction Property of Equality (subtracted 5 from both sides).
Multi-Property Chain
Given: and . Which properties let you conclude ?
Step 1: — Definition of Congruent Segments.
Step 2: and , so — Transitive (or Substitution) Property of Equality.
Answer: Definition of Congruent Segments + Transitive Property of Equality.
Common Pitfalls
Confusing Equality with Congruence
compares numbers (measures). compares figures (segments, angles). Use for measures, for the angles themselves.
Real-Life Applications
Balancing Chemical Equations
Chemists use the same properties of equality to balance equations: if atoms on the left equal atoms on the right, you can add the same compound to both sides (Addition Property) or multiply coefficients (Multiplication Property).
Practice Quiz
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