Answer: no polygon exists in which the ratio of the number of diagonals to the sum of the measures of the angles is 1 to 18, because the number of sides n cannot be equal to 23.
Step-by-step explanation: Let n be the number of sides of the polygon. The number of diagonals in a polygon of n sides is given by the formula:
d = n(n-3)/2
The sum of the measures of the angles in a polygon of n sides is given by the formula:
180(n-2)
The ratio of the number of diagonals to the sum of the measures of the angles is:
d / [180(n-2)] = [n(n-3)/2] / [180(n-2)] = (n-3) / 360
We want to show that this ratio cannot be equal to 1/18, or:
(n-3) / 360 ≠ 1/18
Multiplying both sides by 360, we get:
n-3 ≠ 20
Adding 3 to both sides, we get:
n ≠ 23
Therefore, no polygon exists in which the ratio of the number of diagonals to the sum of the measures of the angles is 1 to 18, because the number of sides n cannot be equal to 23.