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# Help meh

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In a certain pentagon, the interior angles are  a,b,c,d   and e where  a,b,c,d,e are integers strictly less than 180 . ("Strictly less than 180" means they are "less than and not equal to" 180.)

If the median of the interior angles is  61  and there is only one mode, then what are the degree measures of all five angles?

Sep 19, 2017

#1
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In a certain pentagon, the interior angles are  a,b,c,d   and e where  a,b,c,d,e are integers strictly less than 180 . ("Strictly less than 180" means they are "less than and not equal to" 180.)

If the median of the interior angles is  61  and there is only one mode, then what are the degree measures of all five angles?

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The sum of all the interior angles of a pentagon  = 540°

Note that if we require integer solutions....we will have a problem

If the median angle is  61° and we have only a single mode....all the angles must be of  different measure....then..... the largest  possible integer values of the smallest two angles  are 59° and 60°

But note that the sum of the three smallest angles [ 59° + 60° + 61°] = 180°.....so....this means that the remainder of the angles must sum to  540 - 180  = 360°.....but  this would mean that one of the largest two angles will have to be > 180° [because of the single mode requirement] and this isn't allowed   Sep 20, 2017
edited by CPhill  Sep 20, 2017
#2
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actually @CPhill thats wrong

you have 1 mode so you have 3 61s and 178, 179 Guest Sep 21, 2017