Reflexive Property of Equality
Reflexive Property of Equality
The property that a = a. One of the equivalence properties of equality.
See also
Symmetric property of equality, transitive property of equality, transitive property of inequalities
Key Formula
a=a
Where:
- a = Any real number, variable, or expression
Example
Problem: In a two-column proof, you are given a segment AB and need to justify the statement AB = AB. Which property allows you to write this?
Step 1: Identify what the statement claims: the length AB is equal to itself.
AB=AB
Step 2: Recall that the Reflexive Property of Equality states any value equals itself: a=a for all a.
Step 3: Since AB is a real number (a length), it satisfies this property directly.
Answer: The justification is the Reflexive Property of Equality.
Another Example
Problem: Show why the reflexive property is needed to prove that if x + 3 = 7, then 7 = x + 3, using the symmetric property.
Step 1: Start with the given equation.
x+3=7
Step 2: The symmetric property says if a=b, then b=a. But before you can apply the symmetric property, you need the equation to exist — and at its most basic level, you rely on the fact that each side equals itself. The reflexive property guarantees that 7=7 and x+3=x+3 are valid statements.
Step 3: Apply the symmetric property to the given equation to conclude:
7=x+3
Answer: The reflexive property underpins the system of equality that makes applying the symmetric property valid. It ensures every expression has a well-defined equal relationship with itself.
Frequently Asked Questions
Why is the reflexive property of equality important if it's so obvious?
It may seem obvious, but mathematics requires explicit axioms — nothing is assumed without justification. The reflexive property serves as a foundational building block in proofs. Without it, you couldn't formally justify that a quantity equals itself, which is needed as a starting point in many geometric and algebraic proofs.
When do you actually use the reflexive property in a proof?
You use it most often in geometry proofs when two triangles share a common side or angle. For instance, if triangles ABC and DBC share side BC, you write BC=BC by the Reflexive Property to establish one pair of congruent parts for triangle congruence (like SSS or SAS).
Reflexive Property vs. Symmetric Property
The reflexive property says a=a (anything equals itself). The symmetric property says if a=b, then b=a (you can flip the sides of an equation). The reflexive property involves one value; the symmetric property involves two values and a known equation between them. Together with the transitive property, they form the three equivalence properties of equality.
Why It Matters
The reflexive property is essential in geometry proofs, especially when proving triangles congruent. Whenever two figures share a side or angle, you cite the reflexive property to establish that the shared element is congruent to itself. It also serves as one of the three axioms that make equality an equivalence relation, which is the foundation for solving equations and writing valid logical arguments in algebra.
Common Mistakes
Mistake: Confusing the reflexive property with the symmetric property.
Correction: The reflexive property is about one thing equaling itself (a=a). The symmetric property is about reversing an existing equation: if a=b, then b=a. They are different properties.
Mistake: Forgetting to state the reflexive property in a proof when a shared side or angle is used.
Correction: In a two-column proof, every statement needs a reason. When two triangles share a side (like BC=BC), you must explicitly write 'Reflexive Property of Equality' (or 'of Congruence') as the justification. Leaving it out makes the proof incomplete.
Related Terms
- Equivalence Properties of Equality — The three properties that define equality
- Symmetric Property of Equality — If a = b then b = a
- Transitive Property of Equality — If a = b and b = c then a = c
- Transitive Property of Inequalities — Extends transitivity to inequality relations
- Equality — The fundamental relation this property describes
- Properties of Congruence — Parallel reflexive property for geometric congruence