Chapter 7, Problem 30RP

To determine

To show that the number of diagonals in a polygon is never same as the sum of measures of the exterior angles, one per vertex, of the polygon.

Explanation of Solution

The number of diagonals is $\frac{n^2-3n}{2}$

Sum of measures of the exterior angles is $\frac{360}{n}$

Now, they are equal-

  $\begin{array}{l}\frac{n^2-3n}{2}=\frac{360}{n}\\ n^2(n-3)=360\end{array}$

Since n is not an integer, so it can be understood that the sum of measures of the exterior angles and number of the diagonals are not equal.