Question 632046: can you please show to me the proof of the sum of the exterior angles of any polygons which is equal to 720 degrees..thank you.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! The sum of the exterior angles of a polygon is equal to 360 degrees (not 720).
let I = the interior angle of a polygon.
let E = the exterior angle of a polygon.
The exterior angle of a polygon is equal to 180 - the interior angle of a polygon.
This is expressed as:
E = 180 - I
The sum of the interior angles of a polygon is given by the formula:
sum(I) = (n-2)*180 where n is the number of sides of the polygon.
From this formula, the interior angle of a formula is calculated as:
I = (n-2)*180/n
Since the exterior angle of a polygon is always supplementary to the interior angle of a polygon, this means that:
E = 180 - (n-2)*180/n
simplify this formula to get:
E = 180 - (180n-360)/n
Since 180 = 180n/n, this equation can be rewritten as:
E = 180n/n - (180n-360)/n
This can be further simplified to:
E = 180n/n - 180n/n + 360/n
Combine like terms and you get:
E = 360/n
Multiply both sides of this equation by n to get:
n*E = 360
Since n*E is equal to the sum of the exterior angles of any polygon, this means that:
The sum of the exterior angles of any polygon is equal to 360 degrees.
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