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Angles and intersecting lines

## 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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## Sum of the exterior angles of convex polygon

Discussion and questions for this video
Is 360 degrees for all polygons ?
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.........

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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?
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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.
How do you find the exterior angles of a conCAVE polgon, then?
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Actually, it is 360°, according to http://en.wikipedia.org/wiki/Exterior_angle
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Can you have a negative angle?
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Yes, if you say that angles measured counterclockwise are positive and those measured clockwise are negative : )
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In a polygon with n sides at what point does n+1 equal a circle?
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A circle is not a polygon. Polygons are composed of straight sides. A circle does not have straight sides, it has curving sides.
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 circum-circle of a polygon equal a circle? If so, is a circle a polygon?
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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 ).
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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?
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Yes, the answer is 360 degrees and will always be 360 degrees for any convex polygon. Good job on figuring that out!
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is a star considered as a convex polygon?
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Wait a minute... A star is both right? Because it is going inward and outward right?
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what is a convex polygon?
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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.
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What is the meaning of anticlockwise?
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it is the same as counter-clockwise, which is the opposite of the direction the hands of a clock go.

Or if you start at the top of a circle, and go down and around to the left.
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At 3:10 Sal said something like clockwise and counterclockwise..
What does he mean by clockwise and counterclockwise ?
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Clockwise on the clock rotation goes as top-right-down-left.
Counterclockwise on the clock rotation goes as top-left-down-right.

Most clocks go Clockwise.
Why didn't Sal define convex?
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3:29 Oh wait, never mind...

From now on, I think I'll wait until the end of the video before commenting.
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What does he mean by clockwise and counterclockwise ?
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It's the way the hands on an analog clock move. if you have a line segment from the center of a circle going straight up, when it moves to the right going on all the way around however far that is clockwise, if it goes left that is counterclockwise.
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What is the difference between convex and concave shapes ?
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"Convex and Concave Polygons. Every polygon is either convex or concave. The difference between convex and concave polygons lies in the measures of their angles. For a polygon to be convex, all of its interior angles must be less than 180 degrees."
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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?
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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.
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How do you find out the measure of the interior angles with only information about the measure of exterior angles?
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Exterior angles and interior angles are supplementary. Thus
Interior Angle = π - exterior angle (in radians)
Interior Angle = 180° - exterior angle (in degrees)
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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
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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, step-by-step, 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.
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then how do you find the interior angles of a convex polygon?
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To find out only one exterior angle(of a convex polygon), do you do 360/ number of sides?
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No you do not.
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For anyone who is confused. Basically what Sal was trying to say was that every Convex polygon has all of its angles adding up to 360 degrees.
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Sal is *not* saying "that every Convex polygon has all of it angles adding up to 360 degrees." He is saying that the sum of all the *exterior* angles of a convex polygon is 360 degrees. The sum of the interior angles depends on the number of the sides of the polygon. The concept of the sum of the interior angles of a polygon is covered in other videos.
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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?
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What is the sum of the exterior angles of a concave polygon?
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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.
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Are concave polygons also add up to 360 degrees?
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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!
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explain how you can determine if a polygon is concave
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If you can draw diagonals outside a shape, then that shape is a concave polygon.
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I am just curious, but what happens when you have a concave polygon?
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The sum is still 360 but you have to subtract the negative angles caused by the concave parts. It is a bit trickier.
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is A+B+C+D+E+G+H+I= 360 or is it something else?
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The sum of the exterior angles of a polygon with vertices at A,B,C,D,E,F,G,H,I would be 360 degrees. This is a property of the sum of the exterior angles of a polygon.
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is there a difference between the sum of the interior angles of a square and a trapezoid and for all quadrilaterals
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No any quad that isn't concave (dented) would make 2 triangles so it would be 180 * 2 or 360 interior.
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How do you find the exterior angles of a conCAVE polgon, then?
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What previous video is he referencing in the beginning?
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Sum of Interior Angles of a Polygon ?
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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?!
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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.
Why is is that the sum of the exterior angles for all polygons is 360 degrees no matter haw many sides the polygon has?
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Because all a polygon is is a circle with sides. We define a circle as not having any sides, and having its points all equally distanced away from one point. We've given circles 360 degrees to better explain them, and as a result, any closed figures' exterior angles must add to 360.
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what if there is more than six angles?
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When adding up the exterior angles of convex polygons, it doesn't matter how many angles there are. The sum of them is ALWAYS 360 degrees. I'm sorry, I cannot watch the video, because it isn't working for me right now. But by the title I think Mr. Khan was proving this for all polygons. For all convex polygons, the sum of the exterior angles is always 360 degrees.
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At 3:29, doesnt convex have a second meaning?
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Could you find that concave angle by making it into a triangle?
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how do you determine how to solve a regular polygon equation
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what do you mean by solving to find a regular polygon equation?
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convex means what exactly?
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What is a convex region
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I have a question please answer...does this rule of 360 degrees apply to all polygons?
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Yes. It does. Now when you start thinking about regular polygons, really cool patterns start to show up.
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wouldn't an un closed polygon have an exterior angle something different than 360
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a "unclosed polygon" doesn't exsist
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Is the sum of exterior angles 360 for all polygons? And how about triangles?
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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!
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How do you find the perimeter of Convex and Concave polygons?
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what is a convex polygon
?
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Here's a non-rigorous 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
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does a concave polygon also have a measure of 60 degrees?
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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!
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So are all exteriors just 360 and then you divide the shape for interiors with triangles?
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what is the exact equation for exterior angles
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add all the measures of the exterior angles and set them equal to 360
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At 1:39,shouldnt angle A be + to point B?
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Is this true for all polygons that the exterior angles are equal to 360?
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Yes. All exterior angles add up to 360.
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at 1:14 how am i supposed to copy the angle without a protractor? and if i had a protractor, i could easily use it to find the angles without doing this
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wait so why cant it work for concave polygons ?
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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?
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Anything that goes around that doesn't have a beginning or end = 360 degrees.
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how do i algabraically make a hexagon add up to 360?
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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.
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In the video, Sal also did not know about the measurement of any triangle, but he was able to prove it anyways.
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What about the formula: (n-2) x 180?
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It is used to find sum of the interior angles. N = number of sides.
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When you know the number of triangles formed,how do you find the sum of the interior angles of a polygon
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You multiply the number of triangles by 180 and that's the sum of the interior angles.
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I'm still confused as to how to find the sum of all interior angles. Can somebody please explain it in an easier way?
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Are all exterior polygons 360 degrees?
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Yes, the sum of the exterior angles for ANY polygon is 360, making the very interesting formula:
S(Exterior Angles) = 360 degrees`
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What makes a polygon convex?
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in the video the captions are in Chinese ....what do I do?
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wait, so is every polygon's exterior angle is 360 degrees?
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it's applicable only for CONVEX...don't forget that part!
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but what if it is a regular polygon? do you just do the same thing, knowing that exterior angles are the same as the interior angle or is there an another way to find out.
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how did mathmatishins crieit degreases
p.s sorry for my spelling
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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?
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If the sum of interios angles of any polygon is (s-2)*180, then sum of all exterior angles must be 360.Considering that number of exterior angles equal the number of vertices of the polygon, then the sum of exterior angles must be equal to [[180*number of Vertices] - [s-2]*180 (i.e.the sum of the interior angles)].Why does this not work for concave polygons?
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For the concave vertex there isn't an exterior angle in the traditional sense as it's going to be greater than 180 degrees. If you did try and calculate one anyway, you would have a negative angle, at which point the angles would still sum to 360.

This is a good site to see how it works. http://www.mathopenref.com/polygonexteriorangles.html
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What is the difference between concave and convex polygons? (P.s.: did I use the right terms?)
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Concave polygons are polygons that "cave in". If you extend all the sides forever, at least two lines will go through the polygon. Convex polygons are polygons which don't "cave in". If you extend all the sides forever, none of the sides must go through the polygon.
Hope this helps! (P.s: You did!)
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Why doesnt it work for a convave polygon?
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does this apply to all polygons
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If they are convex, yes
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what is the sum of interior convex angles?
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For all polygons would the sum of the exterior angles be 360 degrees or would it be 90 degrees for each exterior angle?
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The exterior angles of any polygon equals 360 degrees.
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can you make a video with how to find the measure of the sides of the parallelograms and rhombus and how to find the angles using their properties
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is the polygon , means more than one sides , am a little confused??
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def Polygon: A closed plane figure bounded by three or more line segments. It comes from the Greek polygonos which means 'many-angled'. So any 2-dimensional object with three or more sides is a polygon (triangles, rectangles, octagons etc).
How would you work out the exterior or interior angle of a concave polygon?
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so how would you find the measure of the exterior angle
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take the interior angle next to and subtract it from 180 or take the 2 opposite interior angle from the exterior angle and add them up
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I'm still terribly confused as to when to use the triangle concept and when to use the circle concept. Sometimes it's hard to tell the difference between concave and convex polygons because the concave polygons are not always "caved" in.
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Polygons are composed of straight sides. A circle does not have straight sides, it has curving sides
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Can a polygon have a higher angle than 360?
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so a concave polygon has one angle greater than 180 degrees?
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If any of the angles on the interior of the polygon are acute, it is concave
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Is it also 360 degrees for a triangle?
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Yes, and for any convex polygon.
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Is geometry part of algebra?
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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.
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Is there some similar rule for finding sum of exterior angles of concave shapes?
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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
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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?
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vertex are points, make those points into angles
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how to you find the exterior angles of a concave polygon ?
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Actually, it's the same as a convex polygon. The sum is also 360°
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a regular polygon has 20 degrees, find the number of sides it has
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I am assuming that the sum of interior angles is 20 degrees.
lets take the number of sides as n
(n-2) * 180 = 20 degrees (sum of interior angles of a polygon having n sides is (n-2) * 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.
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im trying to find the values of the variables for the polygon, triangle.
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What is the major difference between Convex and Concave Polygon?
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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!
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Why don't this method work for Concave Polygon and why it only works for Convex Polygon? What is the reason behind it?
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Why is there no sound in this video!
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your computer is probably messed up
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At 4:35, Sal explains a way that he remembers concave and convex... Is there any other way?
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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).
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The sum of the exterior angles of a polygon, including a regular, is 360 degrees, right? Or is it just for convex polygons?
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Just for convex.
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What is the difference between a convex and concave polygon?
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A convex polygon is any polygon with vertices (corners) pointing outwards. The concave polygon is the opposite. Concave polygons have angles pointing inwards, so it's kind of "caved" in.
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What exactly is a CONVEX polygon?
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Convex polygons Jadi Josh are polygons which are sort of puffed out, they have no indentations or negative angles.

Hope this helps!
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How do you know when to use this strategy and when to use the 180(s-2) strategy from the sum of interior angles of a polygon video?
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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(4-2)
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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?
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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 :)
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Man, I'm learning this stuff in Honors Geometry right now.
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Where is the video that sal referenced at 0:00
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Does this apply to concave polygons ?
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No it does not. The rule that all the exterior angles of a polygon add up to 360 only works on convex polygons.
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