Addition, Subtraction, & Substitution Postulates

In this video you will learn the addition, subtraction, & substitution postulates and how to use them properly in a logic proof. Sometimes the addition, subtraction, & substitution postulates are necessary to prove two angles congruent or two sides congruent.

Let’s first learn what these postulates are:

Addition Postulate: If equal quantities are added to equal quantities, the sums are equal.
For example:
$3+5=5+3$

Substitution Postulate: A quantity may be substituted for its equal in any expression.
For example:
$3+5=5+3$
$8=8$

Since the sum of 3 and 8 are both 8, we can substitute each expression with 8 and they will still equal to one another.

Let’s look at the diagram given in the video:

We are given the information that $\overline{BC} \cong \overline{ED}$ and we have to prove that $\Delta ABC \cong \Delta FCE$ .

Mark the congruent lines on the diagram and then write it in a statement-reason proof.

Note that $\overline{BC}$ and $\overline{ED}$ aren’t sides of the triangles but rather part of the side length. The entire sides are actually $\overline{BD}$ and $\overline{EC}$ .

Both overlap the line segment $\overline{CD}$ . We know that $\overline{CD} \cong \overline{DC}$ because of the reflective property.

Let’s look at the diagram again. Since we already know that $\overline{BC} \cong \overline{ED}$ , therefore $\overline{BC} + \overline{CD} \cong \overline{ED} + \overline{DC}$ because of the addition postulate since the sum of equal quantities added to equal quantities are equal.

We can substitute $\overline{BD} \cong \overline{EC}$ for $\overline{BC} + \overline{CD} \cong \overline{ED} + \overline{DC}$ because of the substitution postulate. Now, we have the two side lengths congruent to each other.

Similar to the addition postulate, we now have a subtraction postulate.

Subtraction Postulate: If equal quantities are subtracted from equal quantities, the differences are equal.
For example:
$9-5=9-5$
$4=4$

Applying the subtraction postulate into a proof, let’s look at another example:

We are given the information $\overline{BD} \cong \overline{EC}$ . Notice that $\overline{BD}$ and $\overline{EC}$ are more than sides of the triangles. However, we can subtract $\overline{CD}$ from each congruent lines segments.

But first, we need to state that $\overline{CD} \cong \overline {DC}$ because of the reflective property. Then, we can say that $\overline{BD} - \overline{CD} \cong \overline{EC} - \overline{DC}$ because of the subtraction postulate since the differences of equal quantities subtracted from equal quantities are equal.

Use the substitution postulate to replace $\overline{BC} - \overline{CD} \cong \overline{ED} - \overline{DC}$ with $\overline{BC}\cong \overline{ED}$ . Now, we have the sides lengths congruent to each other.