Angle Properties of Triangles
Now that we are acquainted with the classifications of triangles, we can begin our extensive study of the angles of triangles. In many cases, we will have to utilize the angle theorems we've seen to help us solve problems and proofs. However, there are some triangle theorems that will be just as essential to know. This first theorem tells us that if we know the measures of two angles of a triangle, it is possible to determine the measure of the third angle.
Triangle Angle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180.
The diagram above illustrates the Triangle Angle Sum Theorem.
Let's do some examples involving the Triangle Sum Theorem to help us see its utility.
(1) Find the measure of ?C.
As with all problems, we must first use the facts that are given to us. Using the diagram, we are given that
Since our goal is to find the measure of ?C, we can use the Triangle Angle Sum Theorem to solve for the missing angle. So we have
Using the angle measures we were given, we can substitute those values into our equation to get.
Having ?C measure out to 26° satisfies the property that the sum of the interior angles of a triangle is 180°.
(2) Find the value of x in the diagram below.
In this exercise, we are given that
Looking at ?RST, we see that two of three angles are given to us. Thus, we can apply the Triangle Angle Sum Theorem to figure out the measure of the third angle:
Note that ?SRT is the vertical angle opposite ?QRP, so we can deduce that
Then, by the definition of congruent angles, we have
Now, we have one of three angle measures of ?QRP. Since we know that m?P = m?Q = x, we can use the Triangle Angle Sum Theorem as follows
We have found the measure of ?P and ?Q to be 67.
In order to comprehend the next theorem, we must learn two more terms that describe angles. The angle formed by one side of a triangle with the extension of another side is called an exterior angle of the triangle.
Exterior angles get their name because they lie on the outsides of triangles.
The two angles that are not adjacent, or next to, the exterior angle of the triangle are called remote interior angles.
Now that we know what these terms mean, we are ready for a theorem that will help us tremendously in our proofs.
Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
Adding the measures of the two remote interior angles of a triangle gives the measure of the exterior angle.
Let's see how the Exterior Angle Theorem can be utilized to help us find the measures of unknown angles in the examples below.
(1) Find the measures of ?1 and ?2 in the figure below.
First, we can solve for m?1 since we are given the measure of two angles of that triangle. This part of the problem is similar to the examples we have already done above. Let's begin with the statements of what we are given, which are:
Now, we can solve for m?1 by using the Triangle Angle Sum Theorem. So we have
In order to solve for the measure of ?2, we will need to apply the Exterior Angle Theorem. We know that the two remote interior angles in the figure are ?S and ?A. Thus, by the Exterior Angle Theorem, the sum of those angles is equal to the measure of the exterior angle. We have
While not always necessary, we can check our solution using our previous knowledge of lines. We see that ?1 and ?2 make up ray AK. And since straight lines have 180° measures, we know that the sum of ?1 and ?2 must be 180. Let's check to make sure:
So, we know we have worked this problem out correctly.
(2) Find m?B.
Let's take a look at the information we have been given first. We know that
Right off the bat, we can apply the Exterior Angle Theorem to help us solve the problem. We have
This does not answer the question, however. The question asked for m?B. The variable x alone does not tell us what the measure of the angle is. So, we must plug x = 4 into our equation for m?B:
Now we have found that the measure of ?B is 39°.