Angles with triangles and polygons
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Sum of the exterior angles of convex polygon
More elegant way to find the sum of the exterior angles of a convex polygon
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 Several videos ago I had a figure that looked something like this, I believe it was a pentagon or a hexagon.
 and what we had to do is figure out the sum of the in particular exterior angles of the hexagon so that
 this angle equaled A, this angle B, C, D and E. The way that we did it the last time we said, well A
 is going to be 180 degrees, minus the interior angle that is supplementary to A, and then we did that
 for each of the angles and then we figured out, we were able to algebraically manipulate it, we were
 able to figure out what the sum of the interior angles were, using... dividing it up into triangles and
 then use that to figure out the exterior angle. So it was a bit of an involved process. But I want to show
 you in this video that there's actually a pretty simple and elegant way to figure out the sum of these
 particular external angles, exterior angles I should say, of this polygon, and it actually works for
 any convex polygon (if you're picking these particular exterior angles I should say) and so the way to
 think about it is you can just redraw the angles. So lets just draw each of them, so let me draw this
 angle right over here, we'll call it angle A or the measure of this angle's A, either way let me draw
 right over here. So this going to be a convex angle right over here
 it's going to have a measure of A, now let me draw angle B, angle B, and i going to draw
 adjacent to angle A, and what you could do is just to think about it
 maybe if we draw a line over here, if we draw a line over here that is parallel to this line
 then the measure over here would also be B,because this is obviously a straight line,
 it would be like transversal, this of course a responding angles, so if u want to draw
 adjacent angle, the adjacent to A, do it like that, or whatever angle this is the measure of B
 and now it is adjacent to A, now let's draw the same thing to C
 We can draw a parallel line to that right over here. And this angle would also be C
 and if we want it to be adjacent to that, we could draw it there, so that angle is C
 C would look something like this, like that then we can move on to D, once again
 we do it in different color, you could do D, right over here or you could shift it over here
 it'll look like that, or shift over here, it'll look like that
 If we just kept thinking of parallel, if all of this line were parallel to each other
 So, let's just draw D like this, so this line is going to parallel to that line
 Finally, you have E, and again u can draw a line that is parallel to this
 right over here and this right over here would be angle E
 or you could draw right over here, right over here
 And when you see it drawn this way, it's clear that when you add up, the
 measure, this angle A,B,C,D and E going all the way around the circle, either way
 it could be going clockwise or it could be counter clockwise but it will going all the way
 around the circle.
 And some of this angle, A+B+C+D+E is just going to be 360 degree
 And this is work for any convex polygon, and when I say convex polygon I mean it is not that dented words
 Just to be clear what I'm talking about, it would work for any convex polygon that is kind of
 I don't want to say regular, regular means it has the same size and angle,
 but it is not dented, this is a convex polygon. This right here is a concave polygon
 Let me draw this, right this way, so this would be a concave polygon
 Let me draw as it having the same number of side, So i just going to dent this two sides
 right very here. Is it right? Let me do the same number sides, So i do that, that, that, that
 and then that's the same side over there, Let me do that and then like that.
 This has 1,2,3,4,5,6, sides and this has 1,2,3,4,5,6 sides. This is concave, sorry
 this is a convex polygon, this is concave polygon, All you have to remember is kind of cave in words
 And so, what we just did is applied to any exterior angle of any convex polygon. I
 Am a bit confused. This applied to any convex polygon and once again if you take this angle
 and added to this angle and added to this angle, this angle, that angle and that angle and I'm not applying that all
 It's going to be the same and I just drew it in that way I could show u
 that they are different angles, i could say that one green, and that one some other colour
 they can all be different but if you shift the angle like this you can see that
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?

Have something that's not a question about this content? 
This discussion area is not meant for answering homework questions.
Is 360 degrees for all polygons ?
The sum of the exterior angles of any convex polygon is 360°
Yes, it is always!
Nope! As Sal told at 3:25 it will work for any Convex Polygon, it is clear theat it is only liable for Convex Polygons. Notice, he didn't said that is for all polygons. So don' t get confused with Convex Polygon and all polygon.........
Hope this will help you out of this!
Hope this will help you out of this!
A basic way of finding the measure of interior angles of a convex polygon: Multiply 180 by the number of sides of the polygon, then subtract 360
Yes, the sum of all exterior angels of a convex polygon is 360 deg, And what's the logic behind this is explained in the video.
"Sum of the exterior angles of CONVEX polygon:"
yes for all pOlygons lol
Imagine that a polygon is nothing more than a incomplete circle.
convex polygons... and triangles (i like triangles!)
Yes the sum of the exterior angles of a CONVEX POLYGON equals 360 degrees
yes it is,as u can see on video
The sum of all external angles on a polygon = 360 degrees. Kind of like in triangles all interior angles = 180 degrees. ;)
for all convex polygons
yes
yes
Yes, but only for convex polygons
Yes that whats my dad said
no, a triangle is 180 degrees, pentagon is 540 degrees, a hexagon is 720 degrees. You can find out how many degrees of a polygon by using this. n as amount of sides.
(n2)*180
(n2)*180
JUST CONVEX POLYGONS!
no. different polygons have different sides . that's according to no. of sides they have.
I was confused by the definition of "exterior angles".
If the interior angle of one corner is, say, 90 degrees (like a corner in a square) then shouldn't the exterior angle be the whole outside of the angle, such as 270? Why is only 90 degrees counted for the exterior angle of a corner instead of 270?
In other words, exterior corners look like they are always greater than 180, but we subtract the 180. Why?
If the interior angle of one corner is, say, 90 degrees (like a corner in a square) then shouldn't the exterior angle be the whole outside of the angle, such as 270? Why is only 90 degrees counted for the exterior angle of a corner instead of 270?
In other words, exterior corners look like they are always greater than 180, but we subtract the 180. Why?
As I think about this now, it makes sense:
You can think of exterior angles as how many degrees you need to turn at each corner if you're tracing the line around the polygon.
When you walk around the block, you turn 90° at each corner, not 270°. The same applies for any shape. If you were to walk around a hexagon, you would turn only 60° at each corner.
You can think of exterior angles as how many degrees you need to turn at each corner if you're tracing the line around the polygon.
When you walk around the block, you turn 90° at each corner, not 270°. The same applies for any shape. If you were to walk around a hexagon, you would turn only 60° at each corner.
It's just the way exterior angles are defined.
From the wikipedia article: "an exterior angle (or external angle) is an angle formed by one side of a simple, closed polygon and a line extended from an adjacent side."
See: http://en.wikipedia.org/wiki/Exterior_angle
From the wikipedia article: "an exterior angle (or external angle) is an angle formed by one side of a simple, closed polygon and a line extended from an adjacent side."
See: http://en.wikipedia.org/wiki/Exterior_angle
because just is that true
How do you find the exterior angles of a conCAVE polgon, then?
Actually, it is 360°, according to http://en.wikipedia.org/wiki/Exterior_angle
The measure of an interior angle plus the measure of its exterior angle is equal to 180 degrees. So if you have a polygon that has in angle that is 200 degrees, what is it's exterior angle? Well let's see, 200 plus what is equal to 180? 20! But negative angles aren't possible...
Using negative exterior angles might get the sum to 360 in some cases, but don't rely on them  they defy logic!
Using negative exterior angles might get the sum to 360 in some cases, but don't rely on them  they defy logic!
What! There are TWO reflex angles for conCAVE polygons.
Can you have a negative angle?
Yes, if you say that angles measured counterclockwise are positive and those measured clockwise are negative : )
No, angles are defined as being between 0 and 360.
is a star considered as a convex polygon?
A star is not a convex polygon. Draw a line between two star points, the line is outside the star.
Wait a minute... A star is both right? Because it is going inward and outward right?
No, it is concave because it has an angle greater than 180 degrees( also known as a reflex angle.
In the example in the video, would the answer be 360 degrees? If so would it always be 360 degrees? If not what would the answer be and how can you find it?
Yes, the answer is 360 degrees and will always be 360 degrees for any convex polygon. Good job on figuring that out!
yes all polygons have an external measure of 360 degrees
In a polygon with n sides at what point does n+1 equal a circle?
A circle is not a polygon. Polygons are composed of straight sides. A circle does not have straight sides, it has curving sides.
what is a convex polygon?
A convex polygon is a polygon in which no line containing a side of the polygon intersects the interior of the polygon.
It's opposite would be a concave polygon, where a line containing one side of the polygon DOES intersect the interior of the polygon.
It's opposite would be a concave polygon, where a line containing one side of the polygon DOES intersect the interior of the polygon.
a concave polygon goes in on itself but a convex polygon doesn't
A convex polygon is a polygon with no angles that have a measure greater than or equal to 180.
The more sides a convex polygon, (with sides equal in length), the more it will look like a circle. Therefore, is a circle a polygon with 360 sides which would be equal to sides/points?
Stated another way 
Since there is no theoretical limit to a polygon at what point does the circumcircle of a polygon equal a circle? If so, is a circle a polygon?
Stated another way 
Since there is no theoretical limit to a polygon at what point does the circumcircle of a polygon equal a circle? If so, is a circle a polygon?
A polygon must have straight sides. It cannot have curving sides. However, as you have shown, a curved side can be made from an infinite amount of straight sides. Combining this together there are only two logical conclusions that I can think of. One, a polygon cannot have an infinite amount of sides. It can have 10 sides. It can have 10 million sides. It can have a million million sides. But it cannot have infinity sides. The other logical conclusion is that infinity sides does not make a curve ( I have no argument to back this up ).
Why didn't Sal define convex?
3:29 Oh wait, never mind...
From now on, I think I'll wait until the end of the video before commenting.
From now on, I think I'll wait until the end of the video before commenting.
yeah man you should wait
How do you find out the measure of the interior angles with only information about the measure of exterior angles?
Exterior angles and interior angles are supplementary. Thus
Interior Angle = π  exterior angle (in radians)
Interior Angle = 180°  exterior angle (in degrees)
Interior Angle = π  exterior angle (in radians)
Interior Angle = 180°  exterior angle (in degrees)
I get that the sum of the exterior angles of a convex polygon add up to 360 degrees but I have a feeling the way Sal explained it in the video is more on the intuitive side. It is not a demonstration of sorts. Is this concept maybe one of those "intuitively obvious" things that cannot really be proved but is taken as given nonetheless?
Well the way Sal explained it does not provide a proof for all convex polygons, or 'the general case'. He just made this to show us the intuitive way of finding the sum of the exterior angles.
However I'm pretty sure there is a way to prove this, though I haven't seen the proof myself. Good question though.
However I'm pretty sure there is a way to prove this, though I haven't seen the proof myself. Good question though.
Yup, haven't seen a "mathematically rigorous" proof either, that's why I was asking. Wasn't sure there is one. :) But I'm guessing I'll find it in some video somewhere as I progress further.
Thanks for the answer!
Thanks for the answer!
then how do you find the interior angles of a convex polygon?
http://www.khanacademy.org/math/geometry/polygonsquadsparallelograms/v/sumofinterioranglesofapolygo this is the video. hope that helps! :)
Watch the video "Sum of the interior angles of a polygon".
we all know that!
Do any of your teachers make you do two column proofs like a chart that seperates and says statement and reason because mine does and its really annoying
I agree that the proofs can be annoying, but I think they are about the most valuable skills that you can learn in geometry. No, you will most likely not need to be able to prove which triangles in a complex drawing are congruent. However, the skill of being able to think your way through a problem, systematically, stepbystep, logically, without making unwarranted assumptions, is a very valuable skill.
So, I'd agree it is annoying at first, but it is a skill worth knowing.
So, I'd agree it is annoying at first, but it is a skill worth knowing.
Yes, when I was in High School we had to do it too. And I agree they are very annoying, but they do lead to a better understanding of geometry.
So for every polygon, even if it had 800 sides, would the exterior angles always add up to 360, assuming it is a convex polygon?
Also, what about a concave polygon? Does the part that is caved inwards have an external angle, although it is over 180 degrees? Or is it just the other exterior angles that are not caved in that add up to 360 degrees?
Also, what about a concave polygon? Does the part that is caved inwards have an external angle, although it is over 180 degrees? Or is it just the other exterior angles that are not caved in that add up to 360 degrees?
Are concave polygons also add up to 360 degrees?
no, the external angles of a concave polygon do not add up to 180... for example, take a 6 sided concave polygon with one internal reflex angle.. the sum of the external angles in this case turns out to be 540=180*5+360*1(4*180) where the (4*180) is sum of internal angles!
How do I find the specific angle of an exterior angle?
em um polígono regular basta dividir 360° pelo numero de lados (360°/n)...
em um polígono irregular prolongue o angulo interno e subtraialo de 180° (angulo interno 180°)
em um polígono irregular prolongue o angulo interno e subtraialo de 180° (angulo interno 180°)
In astronomy, what causes  in the case of regular convex polygons ("regular" convex structures)  the sum of the interior angles to be infinite and the sum of the exterior ones to be constant?!
a Platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex. Five solids meet those criteria, and each is named after its number of faces.
What previous video is he referencing in the beginning?
Sum of Interior Angles of a Polygon ?
Of course he's not talking about exterior because we are watching that right now.
He's talking about Quadrilateral Overview.
He's talking about exterior angles, not interior angles.
explain how you can determine if a polygon is concave
If you can draw diagonals outside a shape, then that shape is a concave polygon.
I do not know what ur talking about, but I know that a property of a parralelagram is that the diagonals congruently bisect each other.
Are all exterior polygons 360 degrees?
Yes, the sum of the exterior angles for ANY polygon is 360, making the very interesting formula:
```S(Exterior Angles) = 360 degrees```
```S(Exterior Angles) = 360 degrees```
What is the difference between a convex and concave polygon?
a regular polygon has 20 degrees, find the number of sides it has
I am assuming that the sum of interior angles is 20 degrees.
lets take the number of sides as n
(n2) * 180 = 20 degrees (sum of interior angles of a polygon having n sides is (n2) * 180)
180*n  360 = 20
n = 19/2
which is not possible so that means it is not a polygon instead for having a polygon the sum of interior angles must be equal or more than 180 degrees.
lets take the number of sides as n
(n2) * 180 = 20 degrees (sum of interior angles of a polygon having n sides is (n2) * 180)
180*n  360 = 20
n = 19/2
which is not possible so that means it is not a polygon instead for having a polygon the sum of interior angles must be equal or more than 180 degrees.
At 4:35, Sal explains a way that he remembers concave and convex... Is there any other way?
In Spanish _con_ means _with_ so concave can mean _with cave_. Another trick to determining concavity is to imagine an elastic band being stretched around the figure, does it touch all sides or does it have to span a gap. If it has to span a gap then the figure must be concave (having a cavity or cave).
Concave can be remembered as the angle 'caving' in.
what is a convex polygon
?
?
Here's a nonrigorous definition: If you can *always* draw a line from any point in the polygon to any other point in the polygon without going outside the polygon, it is convex.
More at Wikipedia: http://en.wikipedia.org/wiki/Convex_polygon
More at Wikipedia: http://en.wikipedia.org/wiki/Convex_polygon
A polygon in which all the interior angles are less than 180 degrees.
does this apply to all polygons
If they are convex, yes
What about the formula: (n2) x 180?
It is used to find sum of the interior angles. N = number of sides.
When I click on these links, it sends me to YouTube. Then, it says I haven't seen the video and I get no credit for it. It used to work before, but not anymore so how do I fix it?
Does this apply to concave polygons ?
No it does not. The rule that all the exterior angles of a polygon add up to 360 only works on convex polygons.
is the polygon , means more than one sides , am a little confused??
def Polygon: A closed plane figure bounded by three or more line segments. It comes from the Greek polygonos which means 'manyangled'. So any 2dimensional object with three or more sides is a polygon (triangles, rectangles, octagons etc).
Does this apply for concave polygons as well?
Yes, however, it sometimes requires using negative angles (the exterior angle of a reflex interior angle is negative).
in the video the captions are in Chinese ....what do I do?
Go on the caption setting and go on english
So are all exteriors just 360 and then you divide the shape for interiors with triangles?
The exterior angles will add to 360 on all CONVEX polygons. If the polygon is CONCAVE, the exterior angles might not add to 360. Sal shows the difference between a convex and concave polygon at 3:50
For all polygons, you can find the sum of the interior angles by multiplying 180 times two less than the number of sides. For example, a square (having four sides) would have an interior angle sum of 180 x (4  2) = 360.
For all polygons, you can find the sum of the interior angles by multiplying 180 times two less than the number of sides. For example, a square (having four sides) would have an interior angle sum of 180 x (4  2) = 360.
What exactly is a CONVEX polygon?
Its the opposite of a CONCAVE polygon
if the measure of an exterior angle of a regular polygon is 45 degrees then the polygon is A. a decagon
B.a an octagon
C.a pentagon
D.a square
B.a an octagon
C.a pentagon
D.a square
Why is there no sound in this video!
your computer is probably messed up
What is the major difference between Convex and Concave Polygon?
A convex polygon can never have an interior angle that is greater than 180 degrees. A concave polygon can. That's the major difference. I hope this helps!
I am slow at math, so bear with me if I get confused but I understand this video in bits and pieces. How do they decide where to draw the lines out to determine the exterior angle of the polygon? Why did he make corresponding angles with parallel lines (transversals) if we're not deducing each angle measurement and only finding out that because of some drawing it will always equal 360 for any convex polygon? Seems a little too indepth if the explanation is really that simple. If it is always 360 degrees, why do we need to know the drawing and adjacent angle explanations? That just threw me off. Can anyone explain that part as well? Thanks in advance.
Why doesnt it work for a convave polygon?
wouldn't an un closed polygon have an exterior angle something different than 360
a "unclosed polygon" doesn't exsist
A polygon is a flat shape consisting of straight lines that are joined to form a closed chain or circuit. Therefore, there is no such thing as an unclosed polygon.
The sum of the exterior angles of a polygon, including a regular, is 360 degrees, right? Or is it just for convex polygons?
How do you know when to use this strategy and when to use the 180(s2) strategy from the sum of interior angles of a polygon video?
I think you can use that strategy all the time with the interior angles of any kind of polygon
example: triangle is 180(3sides 2)
square is 180(42)
example: triangle is 180(3sides 2)
square is 180(42)
at 4:46 you just said it's true about convex polygans,what about cancave polygans?
I just think that it's true about concave polygans too,isn't it?
I just think that it's true about concave polygans too,isn't it?
Where is the video that sal referenced at 0:00
When you know the number of triangles formed,how do you find the sum of the interior angles of a polygon
You multiply the number of triangles by 180 and that's the sum of the interior angles.
At 3:29, doesnt convex have a second meaning?
What is a convex region
how do i algabraically make a hexagon add up to 360?
what is the exact equation for exterior angles
add all the measures of the exterior angles and set them equal to 360
How would you work out the exterior or interior angle of a concave polygon?
This might be a stupid question but how can it equal 360 degrees? a perfect circle is 360 degrees so that would make every polygon a broken circle, but you cant square a circle? So if you have say a square that exterior angles = 360 degrees wouldnt it be able to have the same area as a circle?
Anything that goes around that doesn't have a beginning or end = 360 degrees.
No because it has those extra areas outside the circle. This is a really great question though.
does a concave polygon also have a measure of 60 degrees?
Could you demonstrate that a regular convex polygon with infinite sides has a total sum of exterior angles that is infinite?
Watch the video ; it will help a lot.
Thank you :)
Watch the video ; it will help a lot.
Thank you :)
how do you determine how to solve a regular polygon equation
what do you mean by solving to find a regular polygon equation?
Man, I'm learning this stuff in Honors Geometry right now.
How do you know when you're supposed to add the interior, or the exterior angles. That's the only reason I can't pass this part. Please help!
At 1:39,shouldn`t angle A be + to point B?
Is the sum of exterior angles 360 for all polygons? And how about triangles?
Triangles also have exterior angles of 360 for the sums. An equalateral triangle has 60˚ for each _interior_ angle. That means that each exterior angle _must_ be 120˚. 120*3 is 360. Hope this helped you understand better!
im trying to find the values of the variables for the polygon, triangle.
The polygon exterior angles theorem says that the sum of the measures of the exterior angles of a convex polygon one angle at each vertex is 360. What does one angle at each vertex mean?
vertex are points, make those points into angles
how to you find the exterior angles of a concave polygon ?
Actually, it's the same as a convex polygon. The sum is also 360°
wait, so is every polygon's exterior angle is 360 degrees?
it's applicable only for CONVEX...don't forget that part!
Yes. If the polygon is regular you can also take out the sum of the angles using this property.
Is this true for all polygons that the exterior angles are equal to 360?
Yes. All exterior angles add up to 360.
The sum of the exterior angles of any convex polygon is 360°
Why don't this method work for Concave Polygon and why it only works for Convex Polygon? What is the reason behind it?
so a concave polygon has one angle greater than 180 degrees?
If any of the angles on the interior of the polygon are acute, it is concave
Is there a better way to find out if a polygon is a convex polygon? If we don't know the angles in degrees, we can't be sure that it really does equal 360 degrees, can we? I think I'm confused, haha.
In the video, Sal also did not know about the measurement of any triangle, but he was able to prove it anyways.
Is geometry part of algebra?
Some parts of geometry are a part of algebra, but fir the most part, geometry is geometry. You do use a lot of algebra in geometry in solving segments, tangents and equal angles. So, in a way, algebra s the language of geometry.
I still don't get what convex and concave polygons are....
Convexthis is less than 180 degrees.
Concavethink you are going inside a cave, more than 180 degrees.
Concavethink you are going inside a cave, more than 180 degrees.
If a polygon has 7 edges does that mean that it's 900 degrees?
You figure out the degrees a polygon has by using the formula (n2)180.
When n = the number of edges. Heptagons (7sided polygons), therefore would give the answer of:
(72)180
(5)180
900. So yes, its total number of degrees is 900.
You can then take the equation farther, and divide 900 by the number of sides (in this case, 7) to obtain the degrees of each angle in the heptagon.
900/7 = 128.571... (The formula to find this is (n2)180 / n; n = # of sides.)
When n = the number of edges. Heptagons (7sided polygons), therefore would give the answer of:
(72)180
(5)180
900. So yes, its total number of degrees is 900.
You can then take the equation farther, and divide 900 by the number of sides (in this case, 7) to obtain the degrees of each angle in the heptagon.
900/7 = 128.571... (The formula to find this is (n2)180 / n; n = # of sides.)
yes u can say that
Is 360 degrees for all polygons ?
Yes every exterior angle on a convex polygon addes up to be 360.
no, a triangle is 180 degrees, pentagon is 540 degrees, a hexagon is 720 degrees. You can find out how many degrees of a polygon by using this. n as amount of sides.
(n2)*180.
*edit* But all polygons EXTERIOR angles equal to 360 degrees
(n2)*180.
*edit* But all polygons EXTERIOR angles equal to 360 degrees
ok thank you just finding out the answer of a question someone else posted! ;)
Sorry, Josh, but I think skleefire10 has it backwards. The sum of the INTERIOR angles of any polygon is (number of sides2)*180. The sum of the EXTERIOR angles of any CONVEX polygon is always 360.
On any polygon you can probable extend the lines at any point where two sides meet to have the external a angles equal 360 degrees. Therefore, if you have six points of two lines meeting as shown in the current polygon used, wouldn’t the sum of all the external angles of that polygon be equal to 6 times 360 degrees?
On any polygon you can probable extend the lines at any point where two sides meet to have the external a angles equal 360 degrees. Therefore, if you have six points of two lines meeting as shown in the current polygon used, wouldn’t the sum of all the external angles of that polygon be equal to 6 times 360 degrees?
so a polygons exterior angles always add up to 360?
Yes while the sum of the interior angles is found by taking the # of sides, subtracting 2, and then multiplying by 180 degrees (n2)(180°). It has to be convex in orther for the exterior angles to add up to 360 degrees though.
Yes, from what I and the genius who thought up THIS site( mathisfun.com/geometry ) can tell, They definitely DO.
What would you do for a concave polygon
Can someone please tell me like what the maxium sum of a polygon is? Thanks :)
there is no maximum sum
How do you find out whether the exterior angles are 360 degrees or something else?
In any convex polygon the exterior angles should add up to 360 degrees.
is it always 360?
thank you
For a convex polygon, yes. If we have a concave polygon then it's a whole different matter.
I still don't understand about concave and convex polygons. Can someone please take the time to explain to me?
mdanivas, a.k.a. Pants
mdanivas, a.k.a. Pants
What?
mdanivas, a.k.a. Pants
mdanivas, a.k.a. Pants
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At 2:33, Sal said "single bonds" but meant "covalent bonds."
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