Welcome to a lesson on the interior and exterior angles of a triangle.
The goals are to define interior and exterior angles of a triangle,
and then also state several theorems involving the exterior angles of a triangle.
The interior angles of a triangle are the angles
inside the triangle formed by the sides of the triangle.
So angles one, two, and three
are the interior angles of this triangle.
The vertices are the points where the sides of the triangles meet.
So the vertices are AB and C.
And vertex is singular for vertices. So we have vertex
A, vertex B, and vertex C.
The exterior angles
of a triangle are the angles that form a linear pair
with the interior angles by extending the sides of a triangle.
There are actually two sets of exterior angles
base upon which sides you extend.
What I mean by that, if we take a look at vertex C,
notice the triangle on the left here we extended
side BC through vertex C.
And notice on this triangle here we extended side
AC through vertex C. So if we continue this pattern you can think of these
extensions going counterclockwise,
and these extensions are clockwise.
So again if angles one, two, and three are interior angles,
then angles four, five,
and six would be exterior angles for this triangle.
And notice they form linear pairs with the interior angles.
And looking at this triangle here,
angles four,
five and six would be the exterior angles.
Again they form linear pairs with the interior angles.
It really doesn't matter which sides you extend because
angle four is congruent to angle four, angle five is congruent to angle five,
and angle six is congruent to angle six. Let's take a look at a couple of
example problems.
Here we want to determine the measure of the indicated angles.
So there's a couple of things to remember. First the sum of the interior angles
of a triangle will always be one hundred eighty degrees.
And now we know that the, and we also know that the exterior angles
form a linear pair with the interior angles. So looking at angle two,
we know the measure of angle two plus one hundred thirty-eight degrees
must equal to one hundred eighty degrees because they form a linear pair.
So if we subtract one hundred thirty-eight from both sides of this equation,
we know that the measure of angle two must be forty-two degrees.
And in a similar way,
the measure of angle one plus ninety-four degrees
must equal one hundred eighty degrees.
So if we subtract ninety-four from both sides, the measure of angle one will be
eighty-six degrees.
And now looking at the interior angles,
we know the measure of angle one plus the measure of angle two, plus the measure of angle three
must equal one hundred eighty degrees. Well the measure of angle one is eighty-six degrees,
the measure of angle two is forty-two degrees,
we don't know the measure the angle three, but we know the sum is one hundred eighty degrees.
So eighty-six plus forty-two, that's one hundred twenty-eight.
We subtract one hundred twenty-eight from both sides. We'll have the measure of angle three,
which would give us fifty-two degrees.
And then to determine the measure of angle four we know that angle three and angle four
form a linear pair. So one hundred eighty degrees
minus fifty-two degrees will give us the measure of angle four,
that'll be one hundred twenty-eight degrees.
Let's just take a moment and sum the exterior angles
of this triangle. We have ninety-four degrees,
one hundred twenty-eight degrees, and one hundred thirty-eight degrees.
This is going to be twenty, carry the two,
it's going to be eleven, thirteen, sixteen, carry the one,
three hundred sixty degrees. Notice the interior and exterior angles formed three linear
pairs.
Well three times one hundred eighty degrees would be give hundred forty degrees,
then if we subtract out the sum of the nterior angles, which is always
one hundred eighty degrees, we would get three hundred sixty degrees.
Which matches the sum of our exterior angles.
We'll come back to this idea later on in the video. Let's take a look at another example.
This will go a little bit quicker, we know that the interior and
exterior angles from a linear pair
and therefore, the sum would be one hundred eighty degrees.
So one eighty minus one twenty-four, that's going to give us fifty-six degrees for this angle here.
Over here we'd have one hundred eighty degrees minus ninety-five degrees, that'll be eighty-five degrees.
And then again we know the sum of the interior angle is going to be
one hundred eighty degrees. Fifty-six degrees plus eight-five degrees,
that's gong to be one hundred forty-one degrees.
So one eighty minus one forty-one is going to give us thirty-nine degrees here.
And then angles one and two form a linear pair. So one eighty minus thirty-nine is going to give us
one hundred forty-one degrees for angle two.
Okay let's talk about a couple of theorems that involve exterior angles,
and then we'll prove them in the next couple of videos.
The sum of the exterior angles of any triangle
as well as any polygon is always three hundred sixty degrees.
So again what I was trying to point out earlier is that we know that the
interior and exterior angles
form three linear pairs and therefore, the sum of angles one through six
would be one hundred eighty degrees times three, which is five hundred forty degrees.
And then we also know that the sum of the interior angles of any triangle
is one hundred eighty degrees. So if we take five hundred forty degrees, subtract out
one hundred eighty degrees, this will always be
three hundred sixty degrees. So the sum of
the exterior angles of a triangle will always be
three hundred sixty degrees. And this is also true for any polygon.
Before we talk about the last theorem, let's define remote angles.
The remote angles are the two angles in a triangle
that are not adjacent angles to the specific
exterior angle So if we pick one exterior angle, let's say this angle here,
the two interior angles not adjacent to angle one,
would be angle two an angle three. Angle two and angle three are the remote angles
to angle one.
And these angles have a special relationship. The exterior angle theorem
states that the sum
of the remote interior angles is equal to
the non-adjacent exterior angle. So again, if this is angle one,
this is angle two and three at the remote angles,
the measure of angle two plus the measure of angle three
will equal the measure of angle one. Let's talk about why that is.
If we call this angle four, so the measure of angle one,
plus the measure of angle four equals one hundred eighty degrees.
We also know the sum of the interior angles is one hundred eighty degrees.
So we know that the measure of angle two, plus the measure of angle three,
plus the measure of angle four equals one hundred eighty degrees.
Well if we solve this first equation,
for the measure of angle one. The measure of angle one is one hundred eighty degrees,
minus the measure of angle four.
And if we solve this equation for the measure of angle two plus the measure of angle three,
we would have the same thing. One hundred eighty degrees minus the measure of angle four.
So from this we can conclude that the measure of angle one is equal to the measure of angle
two, plus the measure of angle three.
We'll take a look at a more formal proof of these two theorems in the next two videos.
Thank you for watching.