Welcome to a proof of the sum of the exterior angles of a triangle
is three hundred sixty degrees. Let's talk about our strategy before we write the proof.
Our ulimate goal is to prove that the measure of angle four
plus the measure of angle five, plus the measure of angle six
is equall to three hundred sixty degrees. First thing we should recognize here is that
each pair of interior and exterior angles
form a linear pair. Which means each pair of angles has a sum of
one hundred eighty degrees. So the total sum
of the interior and exterior angles would be one hundred eighty degrees
times three, which is five hundred forty degrees.
But we also know the sum of the interior angles
is one hundred eighty degrees. So if we take the sum of all of the angles,
and then we subtract out the sum of the interior angles, which we know to be
one hundred eighty degrees. It'll give us a sum
for the exterior angles of three hundred sixty degrees,
which is what we're trying to prove. So this gives us the right direction
to construct our proof.
Let's go and take a look. Here we're given triangle ABC.
If we want to prove that the measure of angle four
plus the measure angle five, plus the measure of angle six
equals three hundred sixty degrees. So as usual we'll start by stating the given.
We have triangle ABC.
So for step two, we'll go ahead and list the sum of each linear pair of angles.
So the measure of angle one plus the measure of angle six
equals one hundred eighty degrees.
The measure of angle two
plus the measure of angle five equals one hundred eighty degrees .
And the measure of angle three
plus the measure of angle four equals one hundred eighty degrees.
And the reason for this is the definition
of a linear pair of angles.
So now if we add these equations together,
we can conclude that the measure of angle one plus the measure of angle two,
plus the measure of angle three,
plus the measure of angle four,
plus to measure of angle five,
plus the measure of angle six
equals five hundred forty degrees. And the reason here would just be by the
addition property of equality.
And now what we're going to do is use the triangular sum
theorem and state the measure of angle one
plus the measure angle two, plus the measuring of three
is equal to one hundred eighty degrees.
And again this is the triangular sum theorem.
Well now looking at equations three and four,
if we take equation three and subtract equation four,
we'd be left with just the sum of the exterior angles.
So we'd have the measure of angle four
plus the measure of five, plus the measure of angle six
is going to be equal to five hundred forty degrees
minus one hundred eighty degrees and that'll give us the
three hundred sixty degrees that we need to prove this theorem.
So we took equation three and subtracted equation four,
so our reason is the subtraction property of equality.
And we have our proof.
I hope you found this helpful.