Chapter 12.3, Problem 27E

To determine

To find :themeasure of an interior angle for different number of sides of polygons.

Answer to Problem 27E

a.

x	3	4	5	6	7	8	9	10
y	${60}^{\circ }$	${90}^{\circ }$	${108}^{\circ }$	${120}^{\circ }$	${128.5}^{\circ }$	${135}^{\circ }$	${140}^{\circ }$	${144}^{\circ }$

b. As the number of the sides increases the measure of an interior angle also increases.

c.no, I am not agree because it is less than ${148}^{\circ }$ .

Explanation of Solution

Given : the tableprovided in the question,

Formula used: the measure of an interior angle of a regular n -gon is $\frac{\left(n-2\right)·{180}^{\circ }}{n}$ .

Calculation: to measure the interior angle for different number of sides.

1. Substitute n by according the number of sides,
For n=4,

  $\begin{array}{l}\frac{\left(4-2\right)·{180}^{\circ }}{4}\text{ [substitute n by 4]}\\ \frac{\left(2\right)·{180}^{\circ }}{4}\text{ [subtract]}\\ {90}^{\circ }\text{ [multiply and divide]}\end{array}$

For n=5,

  $\begin{array}{l}\frac{\left(5-2\right)·{180}^{\circ }}{5}\text{ [substitute n by 5]}\\ \frac{\left(3\right)·{180}^{\circ }}{5}\text{ [subtract]}\\ {108}^{\circ }\text{ [multiply and divide]}\end{array}$

For n=6,

  $\begin{array}{l}\frac{\left(6-2\right)·{180}^{\circ }}{6}\text{ [substitute n by 6]}\\ \frac{\left(4\right)·{180}^{\circ }}{6}\text{ [subtract]}\\ {120}^{\circ }\text{ [multiply and divide]}\end{array}$

For n=7

  $\begin{array}{l}\frac{\left(7-2\right)·{180}^{\circ }}{7}\text{ [substitute n by 7]}\\ \frac{\left(5\right)·{180}^{\circ }}{7}\text{ [subtract]}\\ {128.5}^{\circ }\text{ [multiply and divide]}\end{array}$

For n=8

  $\begin{array}{l}\frac{\left(8-2\right)·{180}^{\circ }}{8}\text{ [substitute n by 8]}\\ \frac{\left(6\right)·{180}^{\circ }}{8}\text{ [subtract]}\\ {135}^{\circ }\text{ [multiply and divide]}\end{array}$

For n=9

  $\begin{array}{l}\frac{\left(9-2\right)·{180}^{\circ }}{9}\text{ [substitute n by 9]}\\ \frac{\left(7\right)·{180}^{\circ }}{9}\text{ [ subtract]}\\ {140}^{\circ }\text{ [mltiply and divide]}\end{array}$

For n=10

  $\begin{array}{l}\frac{\left(10-2\right)·{180}^{\circ }}{10}\text{ [substitute n by 10]}\\ \frac{\left(8\right)·{180}^{\circ }}{10}\text{ [subtract]}\\ {144}^{\circ }\text{ [multiply and divide]}\end{array}$

x	3	4	5	6	7	8	9	10
y	${60}^{\circ }$	${90}^{\circ }$	${108}^{\circ }$	${120}^{\circ }$	${128.5}^{\circ }$	${135}^{\circ }$	${140}^{\circ }$	${144}^{\circ }$



2. As the number of the sides increases the measure of an interior angle also increases.
3. When x=11 then,

  $\begin{array}{l}\frac{\left(11-2\right)·{180}^{\circ }}{11}\text{ [substitute n by 11]}\\ \frac{\left(9\right)·{180}^{\circ }}{11}\text{ [subtract]}\\ {147.27}^{\circ }\text{ [multiply and divide]}\end{array}$

Hence,

a.

x	3	4	5	6	7	8	9	10
y	${60}^{\circ }$	${90}^{\circ }$	${108}^{\circ }$	${120}^{\circ }$	${128.5}^{\circ }$	${135}^{\circ }$	${140}^{\circ }$	${144}^{\circ }$

b.As the number of the sides increases the measure of an interior angle also increases.

c. no, I am not agree because it is less than ${148}^{\circ }$ .