Video Transcript
In this video, we will see how we
can complete geometric proofs using the angle sum of a triangle and we’ll also find
the interior and exterior angles of triangles.
Let’s start with the fact that the
sum of the measures of the interior angles in any triangle is 180 degrees. Of course, we’ve probably used this
lots of times in mathematics. But have we ever thought about why
this is true?
In order to see why this is true,
we’ll start with one geometric fact. The angle measures on a straight
line sum to 180 degrees. For example, in this figure, we
could say that 𝑥 degrees plus 𝑦 degrees is equal to 180 degrees. So let’s imagine then that we have
this triangle 𝐴𝐵𝐶 and we want to determine what their angle measures sum to. We can say that these angles are
called 𝑥, 𝑦, and 𝑧. So we want to say, well, what’s 𝑥
degrees plus 𝑦 degrees plus 𝑧 degrees actually equal to?
In order to determine their sum,
we’re going to construct an extra line segment in the figure. We can call this line segment
𝐷𝐸. And the very important thing is, is
that 𝐷𝐸 is parallel to 𝐴𝐵. We want these lines to be parallel
because then we know that we can apply some geometric properties that we know occur
in parallel lines. We know, for example, that
alternate interior angles have equal measure. So this angle at 𝐷𝐶𝐴 will be the
same measure as the angle at vertex 𝐴. It will also be 𝑥 degrees. In the same way, by using the
transversal 𝐶𝐵, we can say that this angle 𝐵𝐶𝐸 must be congruent to the angle
at vertex 𝐵. They’re both equal to 𝑦
degrees.
We notice then that, of course, 𝑥
and 𝑦 and 𝑧 all lie on a straight line. Therefore, their angle sum must be
180 degrees. And these three angles 𝑥, 𝑦, and
𝑧 on the straight line are the same as the three angles of 𝑥, 𝑦, and 𝑧 degrees
within the triangle. And since we have demonstrated this
without using any specific angle measurement, then we can say that this must be true
for all triangles. That is, we have proved this
theorem that we use every day in mathematics that the sum of all the measures of the
interior angles of a triangle is 180 degrees.
We will now see a more specific
example of how we can apply this method to find missing angles in a triangle.
In the given figure, if 𝐴𝐶 is
parallel to 𝐷𝐸, the measure of angle 𝐴𝐵𝐸 equals 55 degrees, and the measure of
angle 𝐶 equals 75 degrees, find the measure of angle 𝐴𝐵𝐶.
The first piece of information that
we are given here is that 𝐴𝐶 is parallel to 𝐷𝐸. We can also fill in the two angle
measures that we are given that the measure of angle 𝐴𝐵𝐸 is 55 degrees and the
measure of angle 𝐶 is equal to 75 degrees. We can then establish that the
angle that we wish to calculate is here, the measure of angle 𝐴𝐵𝐶.
So there are a couple of ways in
which we could find the measure of this unknown angle. But both methods will use the fact
that we have these parallel line segments. By using the parallel lines 𝐴𝐶
and 𝐸𝐷 and the transversal 𝐵𝐶, we can identify a pair of alternate interior
angles. And since alternate interior angles
are congruent, we can say that the measure of angle 𝐷𝐵𝐶 is equal to the measure
of angle 𝐶. These will both be 75 degrees.
We can notice then that these three
angles made at the vertex 𝐵 all lie on a straight line. We recall that the angle measures
on a straight line sum to 180 degrees. Therefore, we can write that the
measure of angle 𝐴𝐵𝐸 plus the measure of angle 𝐴𝐵𝐶 plus the measure of angle
𝐷𝐵𝐶 must be equal to 180 degrees. We can then simply fill in the
angle information that we know. Then, by adding 55 degrees and 75
degrees, we get 130 degrees. We can then simplify this equation
by subtracting 130 degrees from both sides, leaving us with the measure of angle
𝐴𝐵𝐶 is equal to 50 degrees. And so we have found the value of
this unknown angle.
But let’s have a look at the
alternative method that we could have used. Let’s return to the question as we
had it. As previously mentioned, we still
use the parallel lines to help us with this method. But this time, let’s consider the
transversal 𝐴𝐵. Once again, we would use the
property that alternate interior angles are congruent to work out that this angle at
vertex 𝐴 is equal to the measure of angle 𝐴𝐵𝐸. They are both 55 degrees.
However, the geometric property
that we will then apply is that the sum of the measures of the internal angles in a
triangle is 180 degrees. So, if we added 55 degrees, 75
degrees, and the measure of angle 𝐴𝐵𝐶, we would get 180 degrees. And as we have previously worked
out, if we add 55 degrees and 75 degrees and subtract that from 180 degrees, it
leaves us with 50 degrees. Therefore, either method will give
us the answer that the measure of angle 𝐴𝐵𝐶 is 50 degrees.
In the next example, we’ll see a
geometric property that we can prove by using the measures of the interior angles in
a triangle.
In the following triangle 𝐴𝐵𝐶,
if the measure of angle 𝐶 equals the measure of angle 𝐶𝐴𝐷 equals 43 degrees and
the measure of angle 𝐵 equals the measure of angle 𝐵𝐴𝐷, find the measure of
angle 𝐵𝐴𝐶.
We can start this question by
identifying the two pairs of congruent angle measures. We have that the measure of angle
𝐶 is equal to the measure of angle 𝐶𝐴𝐷, and those are both 43 degrees. We also have that the measure of
angle 𝐵 is equal to the measure of angle 𝐵𝐴𝐷, although we aren’t given an exact
measurement for those. We can then identify that the angle
that we wish to calculate is that of the measure of angle 𝐵𝐴𝐶, which occurs at
the vertex 𝐴 in the larger triangle 𝐴𝐵𝐶.
A property that we can apply in
this question is that the sum of the measures of the interior angles in a triangle
is 180 degrees. So then, if we consider the large
triangle 𝐴𝐵𝐶, we can say that the measure of angle 𝐵𝐴𝐶 plus the measure of
angle 𝐵 plus the measure of angle 𝐶 is equal to 180 degrees. From the diagram then, we can
observe that the measure of angle 𝐵𝐴𝐶 actually consists of an angle of 43 degrees
and the measure of angle 𝐵𝐴𝐷.
Then, we are given in the question
that the measure of angle 𝐵 is equal to the measure of angle 𝐵𝐴𝐷. Adding in the measure of angle 𝐶,
which is 43 degrees, we can add the left-hand side, and it will be equal to 180
degrees. We can then simplify this by adding
the two 43 degrees, which is 86 degrees. And we know that there will be two
lots of the measure of angle 𝐵𝐴𝐷. Subtracting 86 degrees from both
sides, we have that two times the measure of angle 𝐵𝐴𝐷 is equal to 94
degrees. Finally, dividing through by two,
we have that the measure of angle 𝐵𝐴𝐷 is 47 degrees. Now that we know the measure of
this angle, we can calculate the measure of angle 𝐵𝐴𝐶. It will be equal to 43 degrees plus
47 degrees, which gives us a final answer of 90 degrees.
We’ll now consider what we mean by
the exterior angle of a triangle. An exterior angle is defined as the
angle formed outside the triangle between any side and the extension of another
side. It’s worth noting that there are
two exterior angles of a triangle at each vertex. For example, if we call this
triangle 𝐴𝐵𝐶, then we could extend either side 𝐴𝐵 or side 𝐵𝐶. However, since both of the exterior
angles at 𝐵 make a straight angle with angle 𝐴𝐵𝐶, then they have equal
measure. Because both exterior angles at a
vertex are congruent, we refer to either of the exterior angles as the exterior
angle. And because the interior and
exterior angles add to 180 degrees, we say that the exterior angle at a vertex of a
triangle is supplementary to the adjacent interior angle.
We’ll now see how we can derive the
next important geometric property about exterior angles. Let’s continue with the triangle
𝐴𝐵𝐶. And we remember that the sum of the
measures of the interior angles of a triangle is 180 degrees. We could say that the measure of
angle 𝐴 plus the measure of angle 𝐵 plus the measure of angle 𝐶 is equal to 180
degrees. And if we label the point on the
line here with 𝐷, we can also say that the measure of angle 𝐶𝐵𝐷 plus the measure
of angle 𝐵 is equal to 180 degrees, because these two angles lie on a straight
line.
If we then return to the first
equation, we can rearrange it to obtain that the measure of angle 𝐵 is equal to 180
degrees subtract the measure of angle 𝐴 plus the measure of angle 𝐶. In the second equation, we can also
make the measure of angle 𝐵 the subject such that this is equal to 180 degrees
subtract the measure of angle 𝐶𝐵𝐷. Then, setting the right-hand side
of both of these equations equal, we have 180 degrees minus the measure of angle 𝐴
plus the measure of angle 𝐶 is equal to 180 degrees minus the measure of angle
𝐶𝐵𝐷. By subtracting 180 degrees from
both sides and then dividing through by negative one, we have that the measure of
angle 𝐴 plus the measure of angle 𝐶 is equal to the measure of angle 𝐶𝐵𝐷.
Let’s now consider what exactly we
have demonstrated with this final equation. Well, we’re really saying that this
exterior angle 𝐶𝐵𝐷 is equal to the sum of the measures of the other two interior
angles in the triangle. And we can formalize this property
as follows. The measure of any exterior angle
of a triangle is equal to the sum of the measures of the two opposite interior
angles in the triangle. We’ll now see an example of how we
can use this property to find the measure of an exterior angle.
In the given figure, find the
measure of the exterior angle, angle 𝐶.
If we look at the diagram, we can
observe that we are given the measures of two of the interior angles of this
triangle 𝐴𝐵𝐶. The measure of angle 𝐴 is 50
degrees, and the measure of angle 𝐵 is 55 degrees. We need to calculate the measure of
this exterior angle at 𝐶, which is given as 𝑥.
The most efficient way to answer
this question is to recall the property that the measure of any exterior angle of a
triangle is equal to the sum of the measures of the opposite interior angles. This means that since 𝐴 and 𝐵 are
the opposite interior angles to the exterior angle at 𝐶, we can say that 𝑥 is
equal to 50 degrees plus 55 degrees. This gives us an answer of 105
degrees.
Alternatively, we could’ve used the
fact that the three interior angles in the triangle add up to 180 degrees. We could therefore write that the
measure of this interior angle 𝐵𝐶𝐴 is equal to 180 degrees subtract 50 degrees
plus 55 degrees. This gives 75 degrees. We would then need to perform
another calculation using the fact that the sum of these two angles at the vertex 𝐶
must be 180 degrees, because they lie on a straight line. So 𝑥 would be equal to 180 degrees
subtract 75 degrees, and that would give us an answer of 105 degrees. Either method gives us the answer
of 105 degrees.
We’ll now see another question,
which will demonstrate the final geometric property that we’ll see in this
video.
What is the sum of the measures of
the exterior angles of a triangle?
In order to consider this question,
let’s sketch a triangle. Here, we have the triangle
𝐴𝐵𝐶. And as we are considering the
exterior angles of the triangle, we can mark these in and label them with 𝑥, 𝑦,
and 𝑧 degrees. And even though we are going to
consider the exterior angles of a triangle, knowing this property about the interior
angles will be useful, that is, that the sum of the measures of the interior angles
of a triangle is 180 degrees.
We could therefore form the
equation that the measure of angle 𝐴 plus the measure of angle 𝐵 plus the measure
of angle 𝐶 is equal to 180 degrees. Then, before we do anything further
with this equation, let’s consider each vertex. Because we know that the exterior
and interior angles at a vertex are supplementary, we could write that 𝑥 degrees
plus the measure of angle 𝐴 is equal to 180 degrees. Likewise, we can say that 𝑦
degrees plus the measure of angle 𝐵 is equal to 180 degrees. And 𝑧 degrees plus the measure of
angle 𝐶 is equal to 180 degrees. We can then rearrange each of these
three equations to make the measure of the interior angle the subject.
We now have these three equations,
which we can substitute in to our first equation. This gives us that 180 degrees
minus 𝑥 degrees plus 180 degrees minus 𝑦 degrees plus 180 degrees minus 𝑧 degrees
is equal to 180 degrees. We can then rearrange this
equation. By subtracting 180 degrees and
adding 𝑥, 𝑦, and 𝑧 to both equations, we have that 360 degrees is equal to 𝑥
plus 𝑦 plus 𝑧. With this equation then, we have
demonstrated that the sum of the three exterior angles of the triangle 𝑥, 𝑦, and
𝑧 degrees is 360 degrees. This question gives us a
demonstration of the very important geometric property that the sum of the measures
of the exterior angles of a triangle is 360 degrees.
We can now finish this video by
summarizing the key points. The sum of the measures of the
interior angles of a triangle is 180 degrees. We saw how we can prove this result
by drawing a line parallel to the base of a triangle through the other vertex and
using the fact that alternate interior angles are congruent. An exterior angle of a triangle is
the angle formed outside the triangle between any side and the extension of another
side. Both exterior angles at a vertex of
a triangle are congruent. An interior angle in a triangle and
an adjacent exterior angle are supplementary. We also saw that the measure of any
exterior angle of a triangle is equal to the sum of the measures of the two opposite
interior angles. And finally, as we saw demonstrated
in the last example, the sum of the measures of the exterior angles in a triangle is
360 degrees.
