Reflexive Property

In algebra, the reflexive property of equality states that a number is always equal to itself.

Reflexive property of equality

If \(a\) is a number, then \(a = a.\)

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 \(\angle A\) is an angle, then \(\angle A \cong \angle A.\)

If \(\overline{AB}\) is a line segment, then \(\overline{AB} \cong \overline{AB}.\)

If \(O\) is a shape, then \(O \cong O.\)

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).

Reflexive property in proofs

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 \(a,\) and \(b\) be numbers such that \(a=b.\) Prove that if \(c\) is a number, then \(ac=bc.\)

Statements Reasons

1. \(a=b\) 1. Given

2. \(ac=ac\) 1. Reflexive property of equality

3. \(ac=bc\) 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 \(\overline{AB} \cong \overline{AD}\) and \(\overline{BC} \cong \overline{CD},\) prove that \(\triangle ABC \cong \triangle ADC.\)

Statements Reasons

1. \(\overline{AB} \cong \overline{AD};\) \(\overline{BC} \cong \overline{CD}\) 1. Given

2. \(\overline{AC} \cong \overline{AC}\) 1. Reflexive property of congruence

3. \(\triangle ABC \cong \triangle ADC\) 3. SSS triangle congruence

Statements	Reasons
1. \(a=b\)	1. Given
2. \(ac=ac\)	1. Reflexive property of equality
3. \(ac=bc\)	3. Substitution property of equality

Statements	Reasons
1. \(\overline{AB} \cong \overline{AD};\) \(\overline{BC} \cong \overline{CD}\)	1. Given
2. \(\overline{AC} \cong \overline{AC}\)	1. Reflexive property of congruence
3. \(\triangle ABC \cong \triangle ADC\)	3. SSS triangle congruence