### Video Transcript

In this video, we will learn how to
find the measure of the exterior angle of a triangle and compare between an exterior
angle and its corresponding remote interior angles. We will begin by defining what we
mean by an exterior and interior angle.

An interior angle is the angle
between adjacent sides of a polygon. An exterior angle is the angle
between one side of a polygon and the extension of an adjacent side. Whilst these definitions are true
for all polygons, in this video, we will only deal with triangles. We can show these angles by
considering the triangle π΄π΅πΆ shown. There are three interior angles at
vertex π΄, vertex π΅, and vertex πΆ. We can show the exterior angle by
extending one of the sides. In this case, we have extended
πΆπ΅. The exterior angle is the angle
made between this extended line and the side length π΄π΅. We will now look at a question that
will lead us to some important properties of interior and exterior angles.

The figure shows the exterior
angles of a given triangle. Find the sum of the exterior
angles.

We recall that an exterior angle of
a triangle lies between one side of the triangle and an extended adjacent side. In the figure drawn, we have a
triangle π΄π΅πΆ. The line π΄π΅ has been extended to
create the exterior angle 108.4 degrees. The side πΆπ΄ has been extended to
create 116.6 degrees. Finally, our third exterior angle
of 135 degrees has been created by extending the side π΅πΆ. We want to calculate the sum of
these three angles. This means that we need to add 135,
116.6, and 108.4. 135 is the same as 135.0.

Adding the numbers in the tenths
column gives us 10, so we carry a one to the ones column. Five plus six plus eight plus one
is equal to 20. We put a zero in the ones column
and carry the two. Adding our numbers in the tens
column gives us six, and adding the numbers in the hundreds column gives us
three. The sum of the exterior angles in
the triangle is 360 degrees. This is true for any triangle. The exterior angles will always sum
to 360 degrees. It is also true for any polygon,
although in this video, we will only deal with triangles.

We now have our first rule when
dealing with exterior and interior angles of a triangle. The exterior angles of any triangle
sum to 360 degrees. If we consider the diagram we saw
earlier of triangle π΄π΅πΆ where line πΆπ΅ has been extended, angle π₯ is an
exterior angle. The interior angle at vertex π΅
lies on a straight line with π₯. This means that the adjacent
interior and exterior angles sum to 180 degrees, as angles on a straight line add up
to 180. The interior angle at vertex π΅
would therefore be equal to 180 minus π₯.

If we let the other two interior
angles be π¦ and π§, this leads us to another property of interior and exterior
angles. We know that the angles in a
triangle sum to 180 degrees. This means that π¦ plus π§ plus 180
minus π₯ is equal to 180. If we add π₯ to both sides of this
equation, we have π¦ plus π§ plus 180 is equal to 180 plus π₯. Subtracting 180 from both sides
gives us π¦ plus π§ is equal to π₯. The sum of the other two interior
angles are equal to the exterior angle. This gives us our third rule. An exterior angle of a triangle is
equal to the sum of the opposite interior angles.

We will now use these three rules
to solve some more problems involving exterior and interior angles.

Is the measure of two right angles
less than, equal to, or greater than the sum of the interior angles of a
triangle?

In order to answer this question,
we need to recall two properties of angles: firstly, the properties of a right angle
and, secondly, the properties of the interior angles of a triangle. We recall that a right angle is
equal to 90 degrees. This means that the measure of two
right angles will be equal to two multiplied by 90 degrees. This is equal to 180 degrees. We also recall that the sum of the
interior angles of a triangle is equal to 180 degrees. As these two angles are the same,
the correct answer is equal to. The measure of two right angles is
equal to the sum of the interior angles of a triangle.

In our next question, we will look
at a special type of triangle.

What is the measure of an exterior
angle of an equilateral triangle?

We recall that the three sides of
an equilateral triangle are equal in length. This means that the three interior
angles at vertexes π΄, π΅, and πΆ are also equal. An exterior angle is created by
extending one of our side lengths, in this case, π΅πΆ. The exterior angle is the angle
between this extended side length and the adjacent side length π΄πΆ. We can now answer this question
using our properties of angles.

The sum of the exterior angles of a
triangle is equal to 360 degrees. As there are three exterior angles,
one way of calculating the measure of one of them would be to divide 360 by
three. 36 divided by three is equal to
12. Therefore, 360 divided by three is
120. The exterior angle of an
equilateral triangle is therefore equal to 120 degrees. An alternative method here would be
to recall that the sum of interior angles of any triangle equals 180 degrees. As our triangle is equilateral, we
know the interior angles are equal, and 180 divided by three is equal to 60. This means that all three interior
angles of an equilateral triangle are 60 degrees.

We know that the exterior angle is
equal to the sum of the two nonadjacent interior angles. As shown in the diagram, we could
add 60 degrees and 60 degrees to give us 120 degrees. This would once again prove that
the exterior angle of an equilateral triangle is 120 degrees. We also notice in this question
that we have a straight line, and the angles on a straight line sum to 180
degrees. The exterior angle plus the
interior angle must add up to 180 degrees.

In our next question, we will
consider different types of angles.

In a triangle πππ, if ππ is
equal to ππ, what type of angle is the exterior angle at vertex π? Is it (A) acute, (B) right, (C)
obtuse, or (D) reflex?

We begin by recalling the
properties of our four angles. An acute angle is less than 90
degrees. A right angle is equal to 90
degrees. An obtuse angle lies between 90
degrees and 180 degrees. Finally, a reflex angle lies
between 180 degrees and 360 degrees. In this question, we are told that
in the triangle πππ, the length of ππ is equal to the length of ππ. This means that we have an
isosceles triangle. In any isosceles triangle, two
interior angles are also equal. In this case, angle πππ is equal
to angle πππ.

We know that angles in a triangle
sum to 180 degrees. This means that the interior angles
at vertex π and vertex π must both be acute. They must both be less than 90
degrees. Otherwise the sum would be greater
than 180. The exterior angle at vertex π is
shown on the diagram. It is the angle between the side
length ππ and the extension of side length ππ. We know that angles on a straight
line sum to 180 degrees. This means that the interior and
exterior angles at vertex π must sum to 180. As the interior angle is acute, it
is less than 90, the exterior angle must be greater than 90 and obtuse. In the isosceles triangle drawn,
the exterior angle at vertex π is obtuse.

In our final question, we will need
to calculate interior and exterior angles of a triangle.

Which of the following is
correct? Is it option (A) the measure of
angle πππ is greater than angle π which is greater than angle πππ? Option (B) angle πππ is greater
than angle πππ which is greater than angle πππ. Option (C) angle π is greater than
angle πππ which is greater than angle πππ. Option (D) angle πππ is greater
than angle πππ which is greater than angle π. Or finally, option (E) angle πππ
is greater than angle πππ which is greater than angle πππ.

As many of the angles in our
options are the same, it makes sense to fill in all the missing angles on the
diagram first. We can do this using our properties
of interior and exterior angles. We know that the angles on a
straight line sum to 180 degrees. We could use this to calculate the
interior angle at vertex π and vertex π. 180 minus 137 is equal to 43. This means that the interior angle
at vertex π is 43 degrees. 180 minus 115 is equal to 65. So the interior angle at vertex π
is 65 degrees.

To calculate the third interior
angle, the one at vertex π, we could use the fact that angles in a triangle sum to
180 degrees. We could add 43 and 65 and then
subtract this from 180. Alternatively, we could use the
fact that any exterior angle of a triangle is equal to the sum of the nonadjacent
interior angles. 137 is therefore equal to 65 plus
π₯. Subtracting 65 from both sides of
this equation gives us π₯ is equal to 72. We could also have calculated this
by subtracting 43 degrees from 115 degrees. The third interior angle is 72
degrees.

We now have the five angles
required. Angle π is equal to 72
degrees. Angle πππ is equal to 65
degrees. Angle πππ is equal to 115
degrees. Angle πππ is equal to 137
degrees. And finally, angle πππ is equal
to 43 degrees. We can now substitute these in to
our five options. All five of our options want the
numbers to be in descending order. This is because the first angle
needs to be greater than the second angle which needs to be greater than the third
angle.

In option (E), 65 is not greater
than 115. In option (D), 43 is not greater
than 65. In option (C), while 72 is greater
than 43, this is not greater than 65. The same problem occurs with option
(B). 115 is greater than 43, but 43 is
not greater than 65. In option (A), our numbers are in
descending order. 137 is greater than 72 which is
greater than 43. This means that the correct answer
is angle πππ is greater than angle π which is greater than angle πππ.

We will now summarize the key
points from this video. We found out in this video that an
exterior angle is the angle between one side of a triangle and the extension of an
adjacent side. An interior angle of a triangle, on
the other hand, lies between two adjacent sides. The angle properties we identified
with regards exterior and interior angles include the following. The exterior angles sum to 360
degrees. The interior angles sum to 180
degrees. We found that any exterior angle of
a triangle is equal to the sum of the two nonadjacent interior angles. Finally, an exterior angle and its
adjacent interior angle sum to 180 degrees.