Question 1087233: three interior angles of a polygon are 160degres each.if the other interior angles are 120degres each,find the number of sides of the polygon?
Found 2 solutions by rothauserc, MathTherapy:
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! each time we add a side to a polygon we add 180 degrees
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for example, 3 sides = 180, 4 sides = 360, 5 sides = 540, 6 sides = 720, ....
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we are given that 3 interior angles are 160 degrees each and the rest are 120 degrees each
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let the sum of the interior angles of our polygon be n, then
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120 must divide (n - 3 * 160)
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we see that a polygon with 6 sides is 720 and (720 - 480) = 240 which is divisible by 120
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furthermore 8 sided polygon is 1080 and (1080 - 480) = 600 which is divisible by 120
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now we can establish a pattern
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Beginning with polygons with 6 sides, we have the sequence 6, 8, 10, 12, ... sided polygons that satisfy the criteria
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Answer by MathTherapy(10557)  (Show Source):
You can put this solution on YOUR website! three interior angles of a polygon are 160degres each.if the other interior angles are 120degres each,find the number of sides of the polygon?
Let the number of sides be n
Sum of INTERIOR angles of ANY POLYGON = 180(n – 2), or 180n - 360
The 3 INTERIOR angles measure 160o each, or sum to 3(160), or 480
The REMAINING number of sides = n – 3
The REMAINING INTERIOR angles measure 120o each, or sum to (n - 3)120, or 120n – 360
Sum of the INTERIOR angles of THIS POLYGON = 480 + 120n – 360
We then get: 180n – 360 = 480 + 120n - 360
180n - 360 = 120n + 120
180n – 120n = 120 + 360
60n = 480
n, or number of sides = 
OR
Let the number of sides be S
Since 3 INTERIOR angles measure 160o each, each corresponding exterior angle measures 20o
Total measure of the 3 corresponding EXTERIOR angles is: 3(20), or 60o
Number of OTHER sides: S - 3
Since the other INTERIOR angles measure 120o each, each of the OTHER corresponding exterior angles measures 60o (180 – 120)
Total measure of the OTHER corresponding EXTERIOR angles is: (S - 3)60, or 60S - 180
With the SUM of the EXTERIOR angles of a polygon being 360o , we get: 60 + 60S – 180 = 360
60S – 120 = 360
60S = 360 + 120
60S = 480
n, or number of sides =
ACCEPT no other answer.
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