Chapter 7.3, Problem 22PSC

a

To determine

To demonstrate: The sum of the measures of the angles of the nonconvex polygon is $540°$

Answer to Problem 22PSC

The sum of the measures of the angles of the nonconvex polygon is $540°$

Explanation of Solution

Given information:

Nonconvex polygon as pentagon

The sum of the measures of the angles of the nonconvex polygon of ‘ n’ sides is given by $\left(n-2\right)\times 180°$

Here in the figure, n = 5

Therefore, The sum of the measures of the angles of the nonconvex polygon will be $\left(5-2\right)\times 180°=540°$

b

To determine

To demonstrate: The sum of the measures of the angles of the nonconvex octagon is $1080°$

Answer to Problem 22PSC

The sum of the measures of the angles of the nonconvex octagon is $1080°$

Explanation of Solution

Given information:

Nonconvex polygon as Octagon

The sum of the measures of the angles of the nonconvex polygon is given by $\left(n-2\right)\times 180°$

Here, n = 8

Therefore, The sum of the measures of the angles of the nonconvex polygon = $\left(8-2\right)\times 180°=6\times 180°=1080°$

c

To determine

To demonstrate: The sum of the measures of the angles of the nonconvex polygon of sides ‘ n’ is $\left(n-2\right)\times 180°$

Answer to Problem 22PSC

Yes, the sum of the measures of the angles of the nonconvex polygon is $\left(n-2\right)\times 180°$

Explanation of Solution

Given information:

The sum of the measures of the angles of the nonconvex polygon of ‘n’ sides is $\left(n-2\right)\times 180°$

Yes, the sum of the measures of the angles of the nonconvex polygon is given by $\left(n-2\right)\times 180°$

d

To determine

To demonstrate: The sum of the measures of the exterior angles of the nonconvex polygon of sides ‘ n’ is $360°$

Answer to Problem 22PSC

The sum of the measures of the exterior angles of the nonconvex polygon of sides ‘ n’ is $360°$

Explanation of Solution

Given information:

The sum of the measures of the exterior angles of the nonconvex polygon of sides ‘ n’ is $360°$

For any closed structure, formed by sides and vertex, the sum of the exterior angles is always equal to the sum of linear pairs and sum of interior angles. Therefore,

  $\begin{array}{l}N=180n–180\left(n-2\right)\hfill \\ N=180n–180n+360\hfill \\ N=360\hfill \end{array}$

Hence, we got the sum of exterior angles of n vertex equal to 360 degrees.