Question

In a certain pentagon, the interior angles are $a^\circ,$ $b^\circ,$ $c^\circ,$ $d^\circ,$ and $e^\circ,$ 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 $61^\circ$ and there is only one mode, then what are the degree measures of all five angles?

Answer 1

The degree measures of all of these angles are 60, 61, 61, 179, 179.

How to solve for the angle measures

Given that the 3rd largest angle that we have here is 61 degrees, the three angles are not going to be more than 183.

The sum of all the angles must be 540 degrees. The total of the largest of the numbers is less that 179 x 2 = 358 degrees.

The sum of the smallest three would be

540 - 358 = 182 degrees

Therefore the possible sets of these angles would be  60, 61, 61, 179, 179 degrees

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