In algebra, the reflexive property of equality states that a number is always equal to itself.
Reflexive property of equality
If is a number, then
In geometry, the reflexive property of congruence states that an angle, line segment, or shape is always congruent to itself.
Reflexive property of congruence
If is an angle, then
If is a line segment, then
If is a shape, then
The reflexive property can seem redundant, but it is used in proofs. It is relevant in proofs because a comparison of a number with itself is not otherwise defined (likewise with a comparison of an angle, line segment, or shape with itself).
The reflexive property can be used to justify algebraic manipulations of equations. For example, the reflexive property helps to justify the multiplication property of equality, which allows one to multiply each side of an equation by the same number.
Let and be numbers such that Prove that if is a number, then
Statements Reasons 1. 1. Given 2. 1. Reflexive property of equality 3. 3. Substitution property of equality
The reflexive property of congruence is often used in geometric proofs when certain congruences need to be established. For example, to prove that two triangles are congruent, 3 congruences need to be established (SSS, SAS, ASA, AAS, or HL properties of congruence). If a side is shared between triangles, then the reflexive property is needed to demonstrate the side's congruence with itself.
Given that and prove that
Statements Reasons 1. 1. Given 2. 1. Reflexive property of congruence 3. 3. SSS triangle congruence