Video Transcript
In this video, we’ll learn how to
find a missing angle’s measure in a quadrilateral, knowing that the sum of angles in
a quadrilateral is 360 degrees. We’ll begin by looking at a short
geometrical proof of this fact before considering how it can help us to solve
problems involving missing angles in quadrilaterals.
We begin our proof with an
axiom. Remember, this is just a statement
or proposition which is accepted as being true. We often use these as a starting
point to a proof. The axiom we’re going to use is
that the sum of the interior angles in a triangle is 180 degrees. Remember, the interior angle is the
angle made by joining two sides of a polygon that sits inside the shape, as shown
here. If we let each of our interior
angles be 𝑥, 𝑦, and 𝑧 degrees, we can then say that 𝑥 plus 𝑦 plus 𝑧 equals 180
degrees.
Now, of course, we’re interested in
the angle sum of a quadrilateral. That’s a polygon with four
sides. So we’re going to add a single
straight line from one vertex to another to split our shape into two triangles. Can you see where this line could
be? Well, to achieve this, we simply
join any vertex to the vertex directly opposite, like this. Notice we’ve now split our
quadrilateral into two triangles. We know that the interior angles in
this triangle sum to 180 degrees. And the interior angles in this
triangle sum to 180 degrees. Then, we see that the interior
angles in our quadrilateral are made up of the combined angles in both
triangles. That means the interior angles in
our quadrilateral sum to 180 plus 180, which equals 360.
And there we have it, a new
fact. The sum of the interior angles in a
quadrilateral of four-sided polygon is 360 degrees. Note though whilst we used a convex
quadrilateral, that is, a four-sided polygon that has interior angles that measure
less than 180 degrees each, it also stands for concave quadrilaterals, such as
arrowheads. And in fact, had we split our shape
up differently, we would have achieved the same result. This geometric proof is actually
hugely powerful for deriving a formula to help us calculate the sum of the interior
angles in any polygon. Whilst it’s outside the scope of
this video to look into this, you might wish to play around with splitting polygons
with a variety of number of sides and see what happens. Let’s now have a look at an
application of the formula.
The sum of the measures of the
angles of a trapezoid is 360 degrees. Write an equation to find the
missing measure 𝑥, and then solve it.
We’re given information about the
sum of the interior angles. But of course, a trapezoid is a
four-sided polygon; it’s a quadrilateral. So we actually know that the
interior angles sum to make 360 degrees. Let’s use this information to write
an equation. We’re going to begin by adding
together the measure of each of our angles. That’s 𝑥 plus 102 plus 116 plus
78. We, of course, know that the sum
total of these four angles has to be 360 degrees. So our equation is 𝑥 plus 102 plus
116 plus 78 equals 360.
We need to solve this to find the
value of 𝑥. So let’s begin by simplifying the
left-hand side. 102 plus 116 plus 78 is 296. So our equation becomes 𝑥 plus 296
equals 360. To solve, we perform the inverse
operations. Well, at the moment, we’re adding
296. So the inverse, the opposite to
that, is to subtract 296. 𝑥 plus 296 minus 296 is just
𝑥. And 360 minus 296 is 64. So the missing measure 𝑥, our
fourth angle at 𝐷, is 64 or 64 degrees.
Now it’s useful to know that with
angle problems, there can be more than one way to solve them. In fact, we recall what we know
about trapezoids. They have a pair of parallel
sides. Our parallel sides have a
transversal; that’s 𝐶𝐷. We know that angles 𝑥 and 116 are
cointerior or supplementary. That means they add to 180
degrees. So the equation we could have
formed would have been 𝑥 plus 116 equals 180. We solve for 𝑥 by subtracting
116. And once again, we find 𝑥 is equal
to 64.
Let’s consider another example.
From the figure, in which the
measure of angle 𝐶𝐷𝐴 is five 𝑥 degrees, the measure of angle 𝐵𝐶𝐷 is seven 𝑦
degrees, and the measure of angle 𝐴𝐵𝐶 is eight 𝑦 degrees, find the values of 𝑥
and 𝑦.
Let’s begin by adding what we’ve
been told to our figure. The measure of angle 𝐶𝐷𝐴 is five
𝑥; that’s this one. The angle 𝐵𝐶𝐷 is seven 𝑦;
that’s this one. And the measure of angle 𝐴𝐵𝐶,
that’s this one, is eight 𝑦. Okay, so how does this help us? Well, we have a couple of useful
facts. Firstly, we know that the sum of
the interior angles in a quadrilateral, that’s a four-sided polygon, is 360
degrees. So according to our diagram, we can
say that eight 𝑦 plus seven 𝑦 plus five 𝑥 plus 85 equals 360. Let’s add seven 𝑦 and eight 𝑦 to
get 15𝑦.
Then we spot that we have two
parallel lines in our diagram. The parallel lines are 𝐵𝐴 and
𝐶𝐷. We can quote that cointerior angles
sum to 180 degrees. So these two angles, 85 and 5𝑥,
are cointerior. They add to 180. And we form a second equation; this
time, 85 plus five 𝑥 equals 180. Let’s solve this equation for
𝑥. We subtract 85 from both sides, so
five 𝑥 is equal to 95. Then we divide through by five. So 𝑥 is 95 divided by five, which
is 19. Remember, we still need to
calculate the value of 𝑦. So let’s clear some space.
We’re going to go back to the first
equation we formed and substitute 𝑥 equals 19. When we do, we get 15𝑦 plus five
times 19 plus 85 equals 360. That’s 15𝑦 plus 95 plus 85 equals
360. 95 plus 85 is 180. So we have an equation in 𝑦 that
we can solve. We subtract 180 from both
sides. So 15𝑦 is 180. And then, we divide through by
15. 𝑦 is 180 divided by 15, which is
equal to 12. So 𝑥 is 19 and 𝑦 is equal to
12.
Now, with angle questions, there’s
often more than one way to answer them. We could have used the fact that
the angles at 𝐵 and 𝐶 are cointerior. They sum to 180 degrees. We form an equation, and we get
eight 𝑦 plus seven 𝑦 equals 180. So 15𝑦 is equal to 180. Once again, we solve this equation
by dividing by 15. And we get 𝑦 is equal to 12.
Let’s now consider another
example.
Given that the measure of angle 𝐷
is two-thirds the measure of angle 𝐵, find the measure of angle 𝐵 and the measure
of angle 𝐷.
To answer this question, we’re
going to begin by adding what we’ve been told to our figure. We’ll define the measure of angle
𝐵 as 𝑥 degrees. So that’s this one. We’re told that the measure of
angle 𝐷 is two-thirds the measure of angle 𝐵. So the measure of angle 𝐷 must be
two-thirds of 𝑥. That’s this one. Okay, so how does that help us? Well, we know that the interior
angles in a quadrilateral, a four-sided polygon, add to make 360 degrees. We can therefore say that
two-thirds 𝑥 plus 53 plus 𝑥 plus 127 must be equal to 360 degrees.
Let’s simplify the left-hand side
of our equation. Firstly, we add 53 and 127 to get
180. Next, we’ll add two-thirds 𝑥 and
𝑥. And we do so by writing 𝑥 as 𝑥
over one and then creating a common denominator of three by multiplying the
numerator and denominator by three. So we get two-thirds 𝑥 plus
three-thirds 𝑥, which is five-thirds 𝑥. So our equation is five-thirds 𝑥
plus 180 equals 360. We solve for 𝑥 by performing a
series of inverse operations.
First, we subtract 180 from both
sides. That leaves us with five-thirds 𝑥
on the left-hand side and 180 on the right. Now, the next thing we could do is
divide through by five-thirds. Alternatively, we can perform this
in two separate steps, the order of which doesn’t actually matter. Let’s multiply both sides of the
equation by three. On the left-hand side, that leaves
us with simply five 𝑥. And on the right-hand side, we get
540. Next, we divide through by five,
giving us 𝑥 on the left-hand side and 108 on the right.
So we found 𝑥 and therefore the
measure of the angle at 𝐵 to be 𝑥 degrees. But what about the measure of the
angle at 𝐷? Remember, we said that that was
two-thirds 𝑥. So if the measure of the angle at
𝐵 is 108, the measure of the angle at 𝐷 is two-thirds of 108 or two-thirds times
108. Well, one-third of 108 is 36, so
two-thirds is double this; it’s 72. And so we find the measure of the
angle at 𝐵 is 108 degrees and the measure of the angle at 𝐷 is 72 degrees. Remember, we could of course check
by adding 108, 72, 53, and 127 and making sure we do indeed get 360 degrees.
In our next example, we’ll look at
how to combine the information about the interior angles of a quadrilateral with
other angle facts.
From the figure, determine the
measure of angle 𝐵𝐸𝐷.
Angle 𝐵𝐸𝐷 is this one. Let’s define it to be equal to 𝑥
degrees. Then, we recall what we actually
know about angles in quadrilaterals. Remember, that’s a four-sided
polygon. We know that the interior angles in
a quadrilateral sum to make 360 degrees. Well, we have a quadrilateral
𝐵𝐶𝐷𝐸. And we know one of its angles is
148 degrees and another is 90. The problem is we can’t find the
measure of 𝐵𝐸𝐷 without knowing a third angle. That’s this one. Let’s call that 𝑦 degrees.
We can work out the measure of
angle 𝑦 or angle 𝐶𝐵𝐸. We know that angles on a straight
line sum to 180 degrees. We see that angle 𝐴𝐵𝐸 is 100
degrees. So 100 plus 𝑦 must be equal to
180. Let’s solve this equation for 𝑦 by
subtracting 100 from both sides. And we see that 𝑦 is 180 minus
100, which is 80. So we now know a third interior
angle in our quadrilateral. Let’s use this information to form
an equation.
The interior angle sum of our
quadrilateral is 80 plus 90 plus 148 plus 𝑥, which is equal to 360. 80 plus 90 plus 148 is 318. So our equation is 318 plus 𝑥
equals 360. We’ll solve this equation for 𝑥 by
subtracting 318 from both sides. 360 minus 318 is 42. So we find 𝑥 is equal to 42. We therefore say that the measure
of angle 𝐵𝐸𝐷 is 42 degrees.
In our very final example, we’ll
look at how we can use ratio to find missing angles in a quadrilateral.
The internal angles of a
quadrilateral are in the ratio three to four to five to six. What is the largest angle?
We’ve been given some information
about the breakdown of four angles in our quadrilateral. So let’s recall what we actually
know about the interior or internal angles of a quadrilateral. We know that they sum to 360
degrees. So what we’re going to need to do
is share 360 degrees into a given ratio. And there are two ways we can do
this. We’ll consider both methods. Let’s consider the first
method.
In our first method, we begin by
adding the ratios together. That’s three plus four plus five
plus six, which is equal to 18. Our second step is to divide our
amount, whatever that is, by this number. Here, that’s to divide 360 by
18. In doing so, we’re finding the size
of one part in our ratio. 36 divided by 18 is two, so 360
divided by 18 is 20. And this means that one part in our
ratio is worth 20 degrees.
The third and final step is to
multiply each part of our ratio by this number. But we’re not going to perform that
fully. We’re just looking to find the size
of the largest angle. In our ratio, that’s represented by
the six. Since we know that one part is
worth 20 degrees, we’re going to multiply six by 20. Six times 20 is 120. And so the largest angle in the
quadrilateral is 120 degrees. Now, what we could do is multiply
the other three parts and check these add to make 360. It’s a good way to check our
answer.
But let’s consider the other
method. The first step in this other method
is the same. We add the ratios. The next step is to form a fraction
for the parts we are interested in. That might be all the parts, but
actually we’re just interested in the largest angle. So that’s the six. Since there are a total of 18
parts, and we’re interested in six of them, the fraction we’re interested in is six
eighteenths. The final step here is to find this
fraction of the amount.
Well, we know we’re sharing 360
degrees, so we’re going to find six eighteenths of 360. But six eighteenths simplifies to
one-third. So we’re going to find one-third of
360. And to find a third, we divide
through by three to get 120. And so, given the information about
the internal angles of our quadrilateral, we once again see that the largest angle
is 120 degrees.
In this video, we’ve learned that
we can split a quadrilateral into two triangles by adding a single straight line
from one vertex to one opposite it to find the sum of its interior angles. When we do, we find that the result
is that the interior angles of a quadrilateral sum to 360 degrees. We can use this fact to solve
problems involving missing angles in quadrilaterals by forming and solving
equations.
