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

In a pentagon, the sum of the interior angles is always 540 degrees. One angle is 61 degrees, and the remaining angles can be found by solving equations based on the sum of angles in a pentagon.

Step-by-step explanation:

In a pentagon, the sum of the interior angles is always equal to $180 imes (5-2) = 540$ degrees.

Let's assume the five angles in the pentagon are $a, b, c, d, e$ in degrees.

The median of the interior angles is $61$ degrees. Since there is only one mode, one of the angles must be $61$ degrees.

So, we have $a, b, c, 61, d$ or $61, a, b, c, d$ as the possible arrangements.

Since the sum of the angles in the pentagon is $540$ degrees, we can set up equations:

For $a, b, c, 61, d$: $a + b + c + 61 + d = 540$

For $61, a, b, c, d$: $61 + a + b + c + d = 540$

We can solve these equations to find the values of $a, b, c,$ and $d$. The remaining angle will be $e = 540 - (a+b+c+d)$.

Answer 2

61, 61, 61, 178, 179

Step-by-step explanation:

The five interior angles of any pentagon add up to 3*180= 540

If the median is 61, then we know the three smallest angles are each no more than 61, and so they add up to at most 3*61= 183 degrees. This means the two largest angles must add up to at least 540-183= 357 degrees. Since 178+179=357, and since the angles are all integers strictly less than 180, the only possibilities for the two largest angles are 178, 179 or 179, 179.

We consider these possibilities one at a time.

If the two largest angles are 178 and 179, then the three smallest angles add up to exactly 183, so they must each be exactly 61. Thus, one set of angles that satisfy the conditions in the problem is 61, 61, 61, 178, 179.

If the two largest angles are 179 and 179, then the three smallest angles add up to 182. The only way this can happen is for one of them to be 60 and the other two to be 61, but in this case, the set of angles would have more than one mode. So we can rule out this answer.

In conclusion, the five angles must be 61, 61, 61, 178, 179.