Question 472301: Question 46 in my book. A polygon has n sides. An interior angle of the polygon and an adjacent exterior angle for a straight angle.
a. What is the sum of the measures of the n straight angles?
b. What is the sum of the measures of the n interior angles?
c. Using your answers abouve, what is the sum of the measures of the n exterior angles?
d. What theorem do the steps above lead to?
I have a feeling you use the Polygon Angle-Sum Theorem: The sum of the measures of the angles of an n-gon is (n-2)180 but I don't know where to use it and also the Polygon Exterior Angle-Sum Theorem: The sum of the measures of the exterior angles of a polygon, one at each vertext, is 360.
Thank you very much! It's much appreciated :-)
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Yes, you do apply those theorems. However I have a feeling you are applying those theorems without enough conceptual understanding of them (e.g. how are they derived?), in which it diminishes the quality of learning.
Deriving the polygon angle-sum theorem is quite straightforward. Given an n-gon, you can always draw n-2 non-overlapping triangles within it (fix a vertex, connect it with two adjacent vertices). The sum of the interior angles in a triangle is 180, so the sum of all the interior angles is 180(n-2).
There are n straight angles in the n-gon so there are 180n degrees total (part a).
Since the interior and exterior angle add up to a straight angle, we subtract the sum of the interior angles from 180n, e.g. 180n - 180(n-2) = 360, the sum of the exterior angles.
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