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Re: Concave Polygons and Exterior/Interior Angles
Posted:
Jul 16, 2005 3:18 PM


Thanks for all of the great responses.
Thank you Kevin for noticing the mistake. I did mean a generalization for the exterior angles of a concave polygon.
These were all great responses, but the reason I am asking is that I am currently working with a company that is developing curriculum for middle/high school students who need to review the basics before jumping into algebra. A part of the curriculum covers basic geometry the students should know, but never quite understood or learned. The point is that this isn't supposed to be a full blown geometry course.
We are trying to state that for any polygon, the sum of the exterior angles is always 360. But we will state this without discussing the difference between a concave or convex polygon.(Because of the limited amount of space.)
At this stage in the program, students have not yet covered negative numbers, so the concept of considering the exterior angles to be negative at concave vertices would seem very foreign to most of them. For that reason, we are afraid of saying this generalization applies to ANY polygon because we will only supply examples of convex polygons.
Does everyone agree that we should just distinguish between concave and convex polygons and THEN talk about the sum of the exterior angles of a concave polygon, and leave the explanation of how it applies to convex polygons to their future geometry teachers?
Thanks again for the help and my apologies for the runon sentences.
ticbol wrote: > What do you mean by generalization? > > As you have said, ".....the sum of the exterior angles for a convex > polygon is 360 degrees", which is true, as long as you measure the > angles in one direction only. Either all clockwise, or all > counterclockwise. > > Is by generalization you mean proof? > There must plenty of proofs for this in the internet. > > One could be, in a convex polygon of n number of sidesan > ngonthere are also n number of straight lines if all the sides are > extended on one end either clockwise or counterclokwise. Each of these > straight lines is an angle of 180 degrees at the point of extension. > Hence, there are a total of n number of 180 degrees fo the whole > ngonin one direction only. > We know that the sum of all of the interior angles of the ngon is > (n2)(180deg). > Then, n(180deg) minus (n2)(180deg) equals the sum of all the exterior > angles. > > n(180) (n2)(180) > = n(180) n(180) +2(180) > = 2(180) > = 360 degrees. > >  > By the way, whether the closed ngon is convex or concave, the sum of > all its interior angles is always (n2)(180) degrees, and the sum of > all its exterior angles in one direction is always 360 degrees. >
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