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:
addition-substitution postulates diagram

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:

subtraction postulate diagram

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.