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### Course: 8th grade > Unit 5

Lesson 2: Triangle angles- Angles in a triangle sum to 180° proof
- Find angles in triangles
- Isosceles & equilateral triangles problems
- Find angles in isosceles triangles
- Triangle exterior angle example
- Worked example: Triangle angles (intersecting lines)
- Worked example: Triangle angles (diagram)
- Finding angle measures using triangles
- Triangle angle challenge problem
- Triangle angle challenge problem 2
- Triangle angles review

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# Triangle angle challenge problem

Learn about the sum of exterior angles in a polygon, specifically a pentagon. We find out that the sum of exterior angles is always the same, no matter the shape of the pentagon. To understand this, we use interior angles and divide the pentagon into triangles. Created by Sal Khan.

## Want to join the conversation?

- Why 3 triangles? Why not 4? 5?(133 votes)
- The number of triangles you should use is (number of sides in polygon)-2. So

5-2=3(168 votes)

- Isn't the sum of the exterior angles for ANY regular polygon always going to be 360 degrees? or am I missing something??(24 votes)
- Actually it is true for ALL polygons, not just regular ones.(38 votes)

- video is not full it ends at6:24(20 votes)
- The same thing happened to me, but all you have to do is go to the far right corner of the video and click on the blue words that say something along the lines of "See full video playback." Hope this helps:)(9 votes)

- very confusing, is there any way someone could explain it easier(9 votes)
- Imagine all of the lines that form the exterior angles extending outward to infinity. Now, imagine zooming out from the pentagon, until it shrinks to a point. You'll see all of the lines that we extended just converging to that point. Now, it's clear that all of those angles form a full circle, which is 360°.

Note, you can try this for any polygon.(22 votes)

- isn't the sum of the angles of a pentagon always 540 degrees though??(9 votes)
- Sum of Interior Angles of a Polygon:
`S(θ) = 180•(n - 2) : where n = number of sides`

The sum of exterior angles of any polygon is always`360º`

(20 votes)

- Did this video end midway through Sal's commentary for anyone else, or is that just my computer's fault?(10 votes)
- The same thing happened to me, but all you have to do is go to the far right corner of the video and click on the blue words that say something along the lines of "See full video playback." Hope this helps:)(4 votes)

- couldnt you just say that some of the angles were ninety and then get your answer from there?(8 votes)
- No, actually, because this information is never technically given. If any of the angles had that 'square' angle box thingy then you could consider those angles 90 degrees, but it's not mathematically legitimate to assume it. Hope it helps.(7 votes)

- Why don't the exterior angles of a polygon always add up to 180? I thought they did...(4 votes)
- Exterior angles of a polygon always add up to 360 degrees, regardeless of the number of sides. You're thinking of interior angles.(12 votes)

- Ok? that seemed easy but when you try it its really really hard(4 votes)
- It's okay. Start with simpler problems first and then gradually increase the level of difficulty once you are confident about the topic.(6 votes)

- Is there a reason it has to be divided into three triangles? If it was an octagon, for example, (or any other shape) how many triangles would it be divided into?(3 votes)
- An octagon would be divided into six triangles. This is something that is much easier to understand by experimenting than through text, but the number of triangles will always be two less than the number of sides. This is why the formula for interior angles of a polygon is 180(n - 2) where n is the number of sides.(5 votes)

## Video transcript

Now this looks like an
interesting problem. We have this polygon. It looks like a pentagon right
over here, has five sides. It's an irregular pentagon. Not all the sides look
to be the same length. And the sides are
kind of continued on. And we have these
particular exterior angles of this pentagon. And what we're asked
is, what is the sum of all of these exterior angles. And it's kind of daunting,
because they don't give us any other information. They don't even give us
any particular angles. They don't start
us off anywhere. And so what we can
do, let's just think about the step by step, just
based on what we do know. Well, we have these
exterior angles. And these exterior
angles, they're each supplementary to
some interior angle. So maybe if we can
express them as a function of the interior angles, we
can maybe write this problem in a way that seems a
little bit more doable. So let's write the
interior angles over here. We already got to e. So let's call this f,
this interior angle f. Let's call this
interior angle g. Let's call this
interior angle h. Let's call this one i. And let's call this one j. And so this sum of these
particular exterior angles, a is now the same
thing as 180 minus g, because a and g
are supplementary. So a is 180 minus g. And then we have plus b. But b we can write in terms
of this interior angle. It's going to be 180 minus h,
because these two angles once again, are supplementary. We do that in a new color. So this is going
to be 180 minus h. And we could do the same
thing for each of them. c, we can write as 180 minus
i, so plus 180 minus i. And then d, we can write as 180
minus j, so plus 180 minus j. And then finally, e, I'm
running out of colors, e, we can write as 180 minus
f, so plus 180 minus f. And so, what we're left with,
if we add up all the 180s, we have 180 5 times. So this is going to be equal to
5 times 180 which is what, 900. And then you have minus g, minus
h, minus i, minus j, minus f. Or we could write
that as minus-- I'll try to do the same
colors-- g plus h-- I'm kind of factoring out
this negative sign-- g plus h-- I'll do
the same color as g, that's not the same color-- g
plus h, plus i, plus j, plus f. And the whole
reason why did this and why this is
interesting now, is that we've expressed
this first thing that we need to figure out. We've expressed it in terms
of sum of the interior angles. So it's going to be 900
minus all of this business. So this is 900 minus all
of this business, which is the sum of the
interior angles. So this is the sum of
the interior angles. So it seems like we've made a
little progress, at least if we can figure out the sum
of the interior angles. And to do that part, I'll
show you a little trick. What you want to do is
divide this polygon, the inside of the
polygon, into three non-overlapping triangles. And so we could do
that from any side. Let's say that
they're all coming out of that side right over there. So there. I have divide it-- let me
do this in a neutral color, I'm doing it in
white-- so that's one triangle right over here. And then let me make another
triangle just like that. So there you go. I've divided into three
non-overlapping triangles. And the reason why I
did that, the reason why this is valuable,
is we know what the sum of the angles
of a triangle add up to. And so to make that
useful, we have to express these angles
in terms of angles that we can figure
out based on the fact that the sums of the
angles, or the measures of the angles in a
triangle add up to 180. So g is kind of already one
of the angles in the triangle. F is made up of two
angles in the triangle. So remember, f is this
entire angle right over here. So let's divide f
into two other angles, or two other measures
of angles, I should say. So let's say that f is equal
to-- we've already gone as high as, let's see, a,
b, c, d, e, f, g, h, i, j-- we haven't used k yet. So let's say that f
is equal to k plus l. It's equal to the sum of
the measures of these two adjacent angles right over here. So f is equal to k plus l. So that way we've
split it up into angles of these other triangles. And then we can do
that with j as well, because j, once again,
is that whole thing. So we could that j is equal to--
let's see, we already used l. So let's say j is
equal to m plus n. And then finally,
we can split up h. h is up here. Remember, it's this whole thing. Let's say that h is the same
thing as o plus p plus q. This is o, this is p, this is q. And once again, I wanted to
split up these interior angles if they're not already
an angle of a triangle. I want to split
them up into angles that are parts of
these triangles. So we have h is equal
to o, plus p, plus q. And the reason why
that's interesting is now we can write the sum
of these interior angles as the sum of a
bunch of angles that are part of these triangles. And then we can use the fact
that, for any one triangle, they add up to 180 degrees. So let's do that. So this expression
right over her is going to be g. g is
that angle right over here. We didn't make
any substitutions. So it's going to
be g-- actually, let me write the whole thing. So we have 900 minus,
and instead of a g, well, actually I'm not
making a substitution, so I can write g plus,
and instead of an h I can write that h
is o plus p plus q. And then plus i. i is sitting right over there. Plus i. And then plus j. And I kind of messed
up the colors. The magenta will go with i. And then j is this
expression right over here. So j is equal to
m plus n instead of writing a j right there. And then finally, we have our f. And f, we've already seen,
is equal to k plus l. So plus k plus l. So once again, I just rewrote
this part right over here, in terms of these
component angles. And now something very
interesting is going to happen, because we know what these
sums are going to be. Because we know that g plus
k plus o is 180 degrees. They are the
measures of the angle for this first
triangle over here, for this triangle
right over here. So g plus o plus
k is 180 degrees. So g-- let me do
this in a new color. So for this triangle
right over here, we know that g plus
o plus k are going to be equal to 180 degrees. So if we cross those out,
we can write 180 instead. And then we also know-- let me
see, I'm definitely running out of colors-- we know that
p, for this middle triangle right over here, we know that
p plus l plus m is 180 degrees. So you take those
out and you know that sum is going to be
equal to 180 degrees. And then finally, this
is the home stretch here. We know that q plus n
plus i is 180 degrees in this last triangle. Those three are also
going to be 180 degrees. And so now we know that the
sum of the interior angles for this irregular
pentagon-- it's actually going to be true for any
pentagon-- is 180 plus 180 plus 180, which is 540 degrees. So that whole thing
is 540 degrees. And if we want to get the
sum of those extra angles, we just subtract it from 900. So 900 minus 540 is
going to be 360 degrees. And we are done. This is equal to 360 degrees.