Geometry: Axioms and Postulates: Axioms of Equality

In this section, we will outline eight of the most basic axioms of equality.

The Reflexive Axiom

The first axiom is called the reflexive axiom or the reflexive property. It states that any quantity is equal to itself. This axiom governs real numbers, but can be interpreted for geometry. Any figure with a measure of some sort is also equal to itself. In other words, segments, angles, and polygons are always equal to themselves. You might think, what else would a figure be equal to if not itself? This is definitely one of the most obvious axioms there is, but it's important nonetheless. Geometric proofs, as well as proofs of all kinds, are so formal that no step goes unwritten. Thus, if perhaps two triangles share a side and you wish to prove those two triangles congruent using the SSS method, it is necessary to cite the reflexive property of segments to conclude that the shared side is equal in both triangles.

The Transitive Axiom

PARGRAPH The second of the basic axioms is the transitive axiom, or transitive property. It states that if two quantities are both equal to a third quantity, then they are equal to each other. This holds true in geometry when dealing with segments, angles, and polygons as well. It is an important way to show equality.

The Substitution Axiom

The third major axiom is the substitution axiom. It states that if two quantities are equal, then one can be replaced by the other in any expression, and the result won't be changed. It seems natural enough, but is necessary to form the foundation of higher math.

The Partition Axiom

The fourth axiom is often called the partition axiom. It states that a quantity is equal to the sum of its parts. Likewise, in geometry, the measure of a segment or an angle is equal to the measures of its parts.

The Addition, Subtraction, Multiplication, and Division Axioms

The last four major axioms of equality have to do with operations between equal quantities.

The addition axiom states that when two equal quantities are added to two more equal quantities, their sums are equal. Thus, if a = b and y = z, then a + y = b + z.
The subtraction axiom states that when two equal quantities are subtracted from two other equal quantities, their differences are equal.
The multiplication axiom states that when two equal quantities are multiplied with two other equal quantities, their products are equal.
The division axioms states axiom states that when two equal quantities are divided from two other equal quantities, their resultants are equal.