In this lesson, you will learn the definition of the Angle Addition Postulate. We will look at some examples so that you understand how this postulate works.
Angle Addition Postulate Defined
The main idea behind the Angle Addition Postulate is that if you place two angles side by side, then the measure of the resulting angle will be equal to the sum of the two original angle measures. For this postulate to apply, the vertices, which are the corner points of the angle, have to also be placed together. We can illustrate this idea by using the heads of two arrows. We are going to label the arrowheads with some points to make it easier to name the angles.
Looking at the outside edges of the two arrowheads, there is a new angle that has been created: angle RLU. This angle has a measure of 80 degrees because it was created by joining a 30-degree angle and a 50-degree angle without any space between them and without overlapping them.
Below, you can see angle RLU with just the shadows of the two added angles in it.
Earlier, we looked at how the Angle Addition Postulate physically combines two angles. In a geometry textbook, you often find the Angle Addition Postulate written like this:If point B lies in the interior of angle AOC, then
Let’s start by looking at the diagram to explain the first part of the theorem.
Angle AOC is created by the two red rays. In the interior of angle AOC is point B. A ray is drawn from point O through B, which splits angle AOC into two parts (angle AOB and angle BOC). The formula in the theorem tells us that if we add the measures of the two parts (angle AOB and angle BOC) together, we get the measure of the big red angle (angle AOC).
Now, let’s look at a couple of examples that apply the Angle Addition Postulate.Example 1: Use the diagram below to find the measure of angle GEM if angle GEO measures 158 degrees and angle MEO is a right angle.
So, let’s think through the Angle Addition Postulate. What is the name of the big angle in the diagram? What are the names of the two smaller angles that combine to create the big angle?Angle GEO is the big angle that is made up of angles GEM and MEO.
So, we can write the formula from the Angle Addition Postulate for these angles:
Angle MEO measures 90 degrees because it is a right angle. Angle GEO measures 158 degrees as was given in the problem statement.
We can substitute in those values into the equation:
Now, we have an equation that we just need to solve for the measure of angle GEM. So, we subtract 90 from both sides. Therefore, angle GEM measures 68 degrees.
Let’s try another example.Example 2: Use the diagram below to find the value of x.
This problem seems to give us a lot less information than the first example because we are not given what any of the angles actually measure. Instead, we are given algebraic expressions for two angles.
However, the diagram gives us additional information that the problem statement did not. Let’s break this problem down the same way we did the first one. What is the name of the big angle? What are the names of the two angles that combine to create the big angle?The two smaller angles are angle ABD and angle DBC. These two angles together form big angle ABC. In the diagram, we can see that angle ABC is a straight angle, which means it measures 180 degrees.
For angle ABD and angle DBC, we are given algebraic expressions for the measures. We can write the formula from the Angle Addition Postulate for these angles and substitute in what we know about each angle’s measure:
Now, we solve the equation. First, we combine the like terms on the left side:
Then we subtract 5 from both sides and divide by 7 to isolate x:
- The Angle Addition Postulate states that: If point B lies in the interior of angle AOC, then
- The postulate describes that putting two angles side by side with their vertices together creates a new angle whose measure equals the sum of the measures of the two original angles.
- A diagram is often helpful for setting up the formula from the Angle Addition Postulate for a particular problem.
The Angle Addition Postulate
- The Angle Addition Postulate states that the measure of an angle formed by two angles side by side is the sum of the measures of the two angles.
- The Angle Addition Postulate can be used to calculate an angle formed by two or more angles or to calculate the measurement of a missing angle.
Upon reaching the end of the lesson, display your ability to:
- State the Angle Addition Postulate
- Write the textbook definition of the postulate
- Use the Angle Addition Postulate to calculate the measure of an angle