Chapter 6.1, Problem 45PPS

To determine

The measure of each interior angle of given regular polygon.

Answer to Problem 45PPS

  $105,110,120,130,135,140,160,170,180,190$

Explanation of Solution

Given:

The measures of interior angles are:

  $x+5,x+10,x+20,x+30,x+35,x+40,x+60,x+70,x+80,x+90$

Calculation:

The sum of the measures of interior angles of an n-sided polygon is:

  $\left(n-2\right)180$

The sides in a decagon are 10.

So, put the value in the formula:

  $\begin{array}{l}Sum=\left(n-2\right)180\\ Sum=\left(10-2\right)180\\ Sum=\left(8\right)180\\ Sum=1440\end{array}$

The sum of the measures of interior angles of a decagon is 1440.

Now,

  $\begin{array}{l}\left[\begin{array}{l}\left(x+5\right)+\left(x+10\right)+\left(x+20\right)+\\ \left(x+30\right)+\left(x+35\right)+\left(x+40\right)+\\ \left(x+60\right)+\left(x+70\right)+\left(x+80\right)+\\ \left(x+90\right)\end{array}\right]=1440\\ 10x+440=1440\\ 10x=1000\\ x=100\end{array}$

Now, use the value of x to calculate the values of all angles:

  $105,110,120,130,135,140,160,170,180,190$