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## Worked examples

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# CA Geometry: Exterior angles

## Video transcript

We're on 41. Lea made two candles in the
shape of right rectangular prisms. So I'm assuming when
they say right rectangular prisms, they mean
a kind of three dimensional rectangular shape. The first candle is 15
centimeters height, 8 centimeters long and
8 centimeters wide. So let's see, it's 15
centimeters high. So that's 15. 8 centimeters long. So maybe that's 8. And 8 centimeters wide. So maybe it goes back 8. So it looks something
like that. That's candle number one. The second candle is 5
centimeters higher, but the same length and width. So the second candle is just
5 centimeters higher. So it looks something
like this. Where this is still 8 and 8. But the height is 5 more
than 15, so it's 20. Fair enough. How much additional
wax was needed to make the taller candle? So if you think about it, we
just have to think about how much incremental volume did we
create by making that section five centimeters higher? So this candle, you can
kind of view it as going up to here. It's 15 centimeters high. And then we added
5 right here. So what's the volume of this
volume right there? So it's 8 by 8 by 5. So 5 times 8 times 8. 5 times 8 is 40 times
8 is 320. So we have to add 320 cubic
centimeters more of wax to make the taller candle. Problem 42. Two angles of a triangle have
measures of 55 and 65. Which of the following could
not be a measure of an exterior angle of
the triangle? So I think this is a good time
to introduce what an exterior angle even is. So if I draw any polygon, and
I'll draw a triangle since that's what this question
is about. So let's say that that's
my triangle. An exterior angle of one
of the vertices is, you essentially extend one of the
lines of the vertices out. So this is an interior
angle right here. The exterior angle is if you
extend this line out, so if I were to draw a dotted
line that extends out this bottom line. This is the exterior
angle right here. As you can see, it's going to
be the supplement to this interior angle. And we could have extended
the line out there. Or we could have extended
this line this way. And we could have
used this one. But we wouldn't add these two
if we wanted to find all of the exterior angles. The exterior angle of this
vertex right here is either this one or this one. And they are the same because
both of these are supplements of this angle. This angle plus either of this
angle or that one will add up to 180 degrees. So that's what an exterior
angle is. So let's go back to
the question. Two angles of a triangle have
measures of 55 and 65. So let's say this is
55 and this is 65. Which of the following could
not be a measure of an exterior angle of
the triangle? Well we can figure out all
of the exterior angles. So first of all what's
this third interior angle going to be? Well they all have
to add up to 180. So let's call that x. So we know that x plus 65
plus 55 is equal to 180. 65 plus 55 is 120. 120 is equal to 180. So x is equal to 60 degrees. So this angle right here, I'll
do it in another color, this is 60 degrees. So what are all the possible
exterior angles. So if I extended this line out
like I did in the example of when I defined what an exterior
angle is, this exterior angle would
be 120 degrees. If were to do it here, if I
would extend this out right here, what would this
exterior angle be? Let's see, this plus
65 is 180. What's 180 minus 65. 180 minus 60 is 120, so this
would have to be 115. So that exterior angle is 115. And then this one, let's
see if I extend it out. One of the two lines that
form the vertex. This is going to be
supplementary to 55. So 180 minus 55 is 125. 180 minus 60 would be 120, and
then it's only 55 so 125. So the three supplementary, or
the three exterior angles of this triangle are 125. And they want to know what
could not be a measure. So 125 is a measure of
an exterior angle. So is 115. And so is 120. So our answer is D. None of the exterior angles
are equal to 130 degrees. Problem 43. OK, they say the sum of the
interior angles of a polygon is the same as the sum of its
exterior angles, what type of polygon is it? And this is an interesting
question. And it's something to experiment
with for yourself. But I want you to draw random
polygons with angle measures, because you know what the angles
all have to add up to in a polyogon. And I think you'll find, that
no matter what polygon you draw, all of the exterior angles
are going to add up to 360 degrees. In fact, in that example we just
did, what were they they were, for that triangle. If I remember, it's
115, 125, and 120. This was for a triangle. If you added them up, you
get 5 plus 5, 10. And then that's 6. 360 degrees. For that triangle, which had
kind of strange angles. It wasn't like an equilateral triangle or anything beautiful. And it's also the same if I
were to draw a rectangle. Well let me not draw
a solid rectangle. So if I have a rectangle
like that. What are the exterior
angles here? Well, I can continue this
line right here. This angle right here
is going to be 90. I could go either way, I could
continue this up, but you can only do it once though for
each of the vertices. Well that exterior
angle is 90. I could go like that, that
exterior angle is 90. I could go like that. That exterior angle is 90. So once again, 90 plus 90 plus
90 plus 90 that's 360 degrees. So it's a good thing to know
that the sum of the exterior angles of any polygon is
actually 360 degrees. And maybe we'll prove that in
another video for a polygon with n sides. But now that we know that, so
they say that the sum of the interior angles of a polygon is
the same as the sum of its exterior angles, this is the
same as saying the sum of the interior angles is
equal to 360. Because this is always going
to be 360 degrees no matter what the polygon is. So they're essentially saying
what polygon's interior angles add up to 360 degrees. And that of course is
a quadrilateral. My mouth got ahead of me. And if you think in a
quadrilateral you have 90, 90, 90, 90 and they add
up to 360 degrees. Next question. Let me copy and paste a couple
of them so I don't have to keep doing this. OK. All right. What is a measure of angle x. So this is an exterior angle
to the vertex B. So how do we figure this out? Well there's kind of a fast
way and a slow way. And the slow way is to figure
out this angle. Because you know that the sum
of the angles add up to 180. And you say oh, x is going
to be 180 minus that. Let's just do it the slow way
and I think you'll see the intuition of a slightly faster
way you could have done it. This plus 60 plus 25
is 85 degrees. Let's call this angle y. So we know that y plus 85
degrees is equal to 180. I got this 85 just by
adding 60 to 25. So this is just saying that
the interior angles of a triangle add up to
180 degrees. And we could figure
out y right now. You could subtract 85 from both
sides and you'd get y is equal to 95. And then we could figure out
x from y, because x is the supplement of y. So then you could say x is
equal to 180 minus 95 and you'd get 85. And that would be fine, it
didn't take you too long, C is the answer. But a slightly faster
way of saying it. OK, y plus 85 is equal to 180. And you also know that y
plus x is equal to 180. So clearly, x is equal to 85. If you add 85 to y, you
get 180, if you add x to y you get 180. So x would be 85. That would be a slightly
faster way of thinking about it. But either way is
fine if you're not under time pressure. OK, problem 45. If the measure of an exterior
angle of a regular polygon. Regular polygon, so that
means that all of the angles are congruent. Of a regular polygon is 120
degrees, how many sides does the polygon have? So this is the vertex in
question, let's say that's the vertex of this polygon
we're thinking about. We want to measure its
exterior angles. So I'd extend one side
of the vertex. And they're saying that
that is 120 degrees. That tells me that the interior
angle at that vertex is 60 degrees. It's the supplement to
the exterior angle. So what regular polygon
has all of its sides equal to 60 degrees? Well, the equilateral
triangle. Regular polygon, all
the angles and all the sides are congruent. So an equilateral triangle
looking something like that would do the trick. It's a regular polygon, all
the sides are the same. And its angles are
60, 60 and 60. So when they say how many sides
does the polygon have? It has three, it's a triangle. I'm out of time. I'll see you in the
next video.