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# hard geometry problem

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The sum of all but one of the angles of a particular convex polygon is 1578 degrees. What is the measure in degrees of the remaining interior angle?

Can someone help, I don't know what to do after making a formula with (n-2)(180). I keep getting irrational degrees.

Dec 6, 2021

#1
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The measure of the remaining interior angle is 27 degrees.

Dec 6, 2021
#2
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Sum of 11 sided poly= (2n-4)x90 deg = 18x90= 1620deg leaves (1620-1578) = 42 deg for last angle

Sum of dodecagon = 1800, leaves 122 for the last angle

Sum of 13 sided poly = 1980, leaves 302 deg for the last angle, BUT since poly is convex, this is impossible.

Hence polygon has 11 or 12 sides, with the last angle being either 42 or 122

Dec 6, 2021
#3
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I think this is how you would do this:

(n-2)(180)  = 1578  +  (n-2)(180) /n

180 n - 360  - 1578  = ( n-2)(180) / n

180n^2 - 1938n = 180 n - 360

180n^2 - 2118 n + 360 = 0               Quadratic formula shows n = 11.59      could be 11 sides or 12 sides

try n = 12         180 (n-2) = 1800     1800 - 1578 = 222  for remaining side     (too big....must be less than 180)

try n = 11         180 ( n-2) = 1620    1620 - 1578 =  42 degrees    OK

Dec 6, 2021