Angles and intersecting lines

# Angles with triangles and polygons

## 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.........

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?
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 : )
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.
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?
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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is a star considered as a convex polygon?
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!
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.
What is the meaning of anticlockwise?
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 ?
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?
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 ?
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.
Can a polygon have a higher angle than 360?
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This can be answered by my program that I made just for this question, here,

How do you find out the measure of the interior angles with only information about the measure of exterior angles?
This can be answered by my program that I made just for this question, here,

What is the difference between convex and concave shapes ?
This can be answered by my program that I made just for this question, here,

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.
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, 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.
then how do you find the interior angles of a convex polygon?
To find out only one exterior angle(of a convex polygon), do you do 360/ number of sides?
This can be answered by my program that I made just for this question, here,

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.
This can be answered by my program that I made just for this question, here,

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?
This can be answered by my program that I made just for this question, here,

What is a convex polygon?
A convex polygon is a polygon that is not caved in. Have you ever seen an arrow that looks like this: ➢? That is a concave polygon. This:∇ is a convex polygon. If you still don't "get it" I would look at this link for more information (and pictures) because this is kind of hard to explain. http://en.wikipedia.org/wiki/Convex_and_concave_polygons
Hope this helps!
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What is the sum of the exterior angles of a concave polygon?
This can be answered by my program that I made just for this question, here,

Why isn't an exterior angle the angle that goes all the way around the corner, greater than 180?
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.
This can be answered by my program that I made just for this question, here,

This can be answered by my program that I made just for this question, here,

Are concave polygons also add up to 360 degrees?
This can be answered by my program that I made just for this question, here,

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 am just curious, but what happens when you have a concave polygon?
The sum is still 360 but you have to subtract the negative angles caused by the concave parts. It is a bit trickier.
is A+B+C+D+E+G+H+I= 360 or is it something else?
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
No any quad that isn't concave (dented) would make 2 triangles so it would be 180 * 2 or 360 interior.
Can't you just "pop out" concave angles to make a convex polygon? The change of the angle would be the same, right?
No, because the two angles formed by the caved in part of a concave polygon wouldn't be there anymore, and the convex polygon wouldn't be congruent to the concave polygon. I hope this makes sense! :D
How do you find the exterior angles of a conCAVE polgon, then?
What previous video is he referencing in the beginning?
Sum of Interior Angles of a Polygon ?
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.
Why is is that the sum of the exterior angles for all polygons is 360 degrees no matter haw many sides the polygon has?
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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HELP! The exercises *and* the video aren't very explanatory and I' m confused in the exercise "Angles of a polygons".
No matter what type of polygon you have, the sum of the exterior angles is ALWAYS equal to 360°. If you are working with a regular polygon, you can determine the size of EACH exterior angle by simply dividing the sum, 360, by the number of angles. Remember, this will ONLY work in a regular polygon (CONVEX).
http://www.regentsprep.org/regents/math/geometry/gg3/lpoly3.htm
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Would students benefit from learning that the sum of exterior angles is tau radians?
what if there is more than six angles?
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.
At 3:29, doesnt convex have a second meaning?
Could you find that concave angle by making it into a triangle?
how do you determine how to solve a regular polygon equation
what do you mean by solving to find a regular polygon equation?
convex means what exactly?
What is a convex region
wouldn't an un closed polygon have an exterior angle something different than 360
a "unclosed polygon" doesn't exsist
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!
How do you find the perimeter of Convex and Concave polygons?
what is a convex polygon
?
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
does a concave polygon also have a measure of 60 degrees?
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!
So are all exteriors just 360 and then you divide the shape for interiors with triangles?
what is the exact equation for exterior angles
add all the measures of the exterior angles and set them equal to 360
How do you find the sum of the exterior angles of a concave polygon?
They still add up to 180˚ provided that you remember to add up the "negative" angles properly.
I can't find the video mentioned in this one about finding exterior angles the other way. Only the sum of interior angles of a polygon video shows up under the "need more help? Watch this video" area and I can't seem to find the video on the side bar thingy (don't know what to call it) The angles of a polygon practice includes exterior angles and that's the practice I'm working on right now. Please help! Thank you.
At 1:39,shouldn`t angle A be + to point B?
Is this true for all polygons that the exterior angles are equal to 360?
Yes. All exterior angles add up to 360.
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
I am still a little confused. What are convex and concave polygons?
A concave surface curves inward. The word is easy to remember because a concave indentation in a wall makes a cave. A convex surface curves outward. Like many pairs of antonyms that are relatively rare and similar in sound, these two adjectives are easy to confuse.
Concave and convex are also geometrical terms; a concave polygon has at least one angle greater than 180 degrees, and a convex polygon is made of angles each less than or equal to 180 degrees.
http://grammarist.com/usage/concave-convex/
wait so why cant it work for concave polygons ?
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.