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:

**Substitution Postulate:** A quantity may be substituted for its equal in any expression.

For example:

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 and we have to prove that .

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

Note that and aren’t sides of the triangles but rather part of the side length. The entire sides are actually and .

Both overlap the line segment . We know that because of the reflective property.

Let’s look at the diagram again. Since we already know that , therefore because of the addition postulate since the sum of equal quantities added to equal quantities are equal.

We can substitute for 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:

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

We are given the information . Notice that and are more than sides of the triangles. However, we can subtract from each congruent lines segments.

But first, we need to state that because of the reflective property. Then, we can say that because of the subtraction postulate since the differences of equal quantities subtracted from equal quantities are equal.

Use the substitution postulate to replace with . Now, we have the sides lengths congruent to each other.