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.
Examples
(1) Find the measure of ?C.
Solution:
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.
Solution:
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.
Examples
(1) Find the measures of ?1 and ?2 in the figure below.
Solution:
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.
Solution:
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°.