Answer:We can use equations to represent the measures of the angles described above. One equation might tell us the sum of the angles of the triangle. For example,
x + y + z = 180
We know this is true, because the sum of the angles inside a triangle is always 180 degrees. What is w? We don't know yet. But, we may observe that the measure of angle w plus the measure of angle z = 180 degrees, because they are a pair of supplementary angles. Notice how Z and W together make a straight line? That's 180 degrees. So, we can make a new equation:
w + z = 180
Then, if we combine the two equations above, we can determine that the measure of angle w = x + y. Here's how to do that:
x + y + z = 180 (this is the first equation)
w + z = 180 (this is the second equation)
Now, rewrite the second equation as z = 180 - w and substitute that for z in the first equation:
x + y + (180 - w) = 180
x + y - w = 0
x + y = w
Interesting. This tells us that the measure of the exterior angle equals the total of the other two interior angles. In fact, there is a theorem called the Exterior Angle Theorem which further explores this relationship:
Exterior Angle Theorem
The measure of an exterior angle (our w) of a triangle equals to the sum of the measures of the two remote interior angles (our x and y) of the triangle.
Let's try two example problems.
Example A:
If the measure of the exterior angle is (3x - 10) degrees, and the measure of the two remote interior angles are 25 degrees and (x + 15) degrees, find x.
First example of finding an exterior angle
To solve, we use the fact that W = X + Y. Note that here I'm referring to the angles W, X, and Y as shown in the first image of this lesson. Their names are not important. What is important is that an exterior angle equals the sum of the remote interior angles.
, where
be any real constant.
such that
