Chapter 12, Problem 19CR

a.

To determine

To calculate: The measure of angle 1 in triangle.

Answer to Problem 19CR

The measure of angle 1 is ${60}^{\circ }$ .

Explanation of Solution

Given information:

Triangle is regular.

Calculation:

  

If the triangle shown is regular then it is also equiangular.

All angles in that triangle are ${60}^{\circ }$ .

  $\angle 1={60}^{\circ }$

b.

To determine

To find: The measure of angle 2 in hexagon.

Answer to Problem 19CR

The measure of angle 2 is ${135}^{\circ }$ .

Explanation of Solution

Given information:

Hexagon is regular.

Formula used:

Sum of interior angle of a polygon:

  $S_i=\left(n-2\right)\left({180}^{\circ }\right)$

n = number of sides

Calculation:

  

  $\begin{array}{l}n=8\\ S_i=\left(n-2\right)\left({180}^{\circ }\right)\\ S_i=\left(8-2\right)\left({180}^{\circ }\right)\\ S_i=6\left({180}^{\circ }\right)\\ S_i={1080}^{\circ }\\ \angle 2=\frac{S_i}{n}=\frac{{1080}^{\circ }}{8}={135}^{\circ }\end{array}$

c.

To determine

To find: The measure of angle 3 in octagon.

Answer to Problem 19CR

The measure of angle 3 is ${120}^{\circ }$ .

Explanation of Solution

Given information:

Octagon is regular.

Formula used:

Sum of interior angle of a polygon:

  $S_i=\left(n-2\right)\left({180}^{\circ }\right)$

n = number of sides

Calculation:

  

  $\begin{array}{l}n=6\\ S_i=\left(n-2\right)\left({180}^{\circ }\right)\\ S_i=\left(6-2\right)\left({180}^{\circ }\right)\\ S_i=4\left({180}^{\circ }\right)\\ S_i={720}^{\circ }\\ \angle 3=\frac{S_i}{n}=\frac{{720}^{\circ }}{6}={120}^{\circ }\end{array}$

c.

To determine

To find: The measure of angle 4.

Answer to Problem 19CR

The measure of angle 4 is ${45}^{\circ }$ .

Explanation of Solution

Given information:

Each polygon is regular.

Formula used:

Sum of interior angle of a polygon:

  $S_i=\left(n-2\right)\left({180}^{\circ }\right)$

n = number of sides

Calculation:

  

  $\angle 1={60}^{\circ }$

  $\begin{array}{l}n=8\\ S_i=\left(n-2\right)\left({180}^{\circ }\right)\\ S_i=\left(8-2\right)\left({180}^{\circ }\right)\\ S_i=6\left({180}^{\circ }\right)\\ S_i={1080}^{\circ }\\ \angle 2=\frac{S_i}{n}=\frac{{1080}^{\circ }}{8}={135}^{\circ }\end{array}$

  $\begin{array}{l}n=6\\ S_i=\left(n-2\right)\left({180}^{\circ }\right)\\ S_i=\left(6-2\right)\left({180}^{\circ }\right)\\ S_i=4\left({180}^{\circ }\right)\\ S_i={720}^{\circ }\\ \angle 3=\frac{S_i}{n}=\frac{{720}^{\circ }}{6}={120}^{\circ }\end{array}$

The measure of angle 4 is

  $\begin{array}{l}\angle 4={360}^{\circ }-(Angleoftriangle+Angleofhexagon+Angleofoctagon)\\ \angle 4={360}^{\circ }-(\angle 1+\angle 3+\angle 2)\\ \angle 4={360}^{\circ }-({60}^{\circ }+{120}^{\circ }+135^{\circ })\\ \angle 4={360}^{\circ }-315^{\circ }\\ \angle 4=45^{\circ }\end{array}$

d.

To determine

To find: A regular pentagon will fit at angle 5.

Answer to Problem 19CR

No

Explanation of Solution

Given information:

Each polygon is regular.

Formula used:

Sum of interior angle of a polygon:

  $S_i=\left(n-2\right)\left({180}^{\circ }\right)$

n = number of sides

  

  $\angle 1={60}^{\circ }$

  $\begin{array}{l}n=8\\ S_i=\left(n-2\right)\left({180}^{\circ }\right)\\ S_i=\left(8-2\right)\left({180}^{\circ }\right)\\ S_i=6\left({180}^{\circ }\right)\\ S_i={1080}^{\circ }\\ \angle 2=\frac{S_i}{n}=\frac{{1080}^{\circ }}{8}={135}^{\circ }\end{array}$

  $\begin{array}{l}n=6\\ S_i=\left(n-2\right)\left({180}^{\circ }\right)\\ S_i=\left(6-2\right)\left({180}^{\circ }\right)\\ S_i=4\left({180}^{\circ }\right)\\ S_i={720}^{\circ }\\ \angle 3=\frac{S_i}{n}=\frac{{720}^{\circ }}{6}={120}^{\circ }\end{array}$

The interior angle of a regular pentagon:

  $\begin{array}{l}n=5\\ S_i=\left(n-2\right)\left({180}^{\circ }\right)\\ S_i=\left(5-2\right)\left({180}^{\circ }\right)\\ S_i=3\left({180}^{\circ }\right)\\ S_i={540}^{\circ }\\ \angle =\frac{S_i}{n}=\frac{{540}^{\circ }}{5}={108}^{\circ }\end{array}$

The measure of angle 5 is

  $\begin{array}{l}\angle 5+Angleofhexagon+Angleofoctagon=3{60}^{\circ }\\ {120}^{\circ }+\angle 5+{135}^{\circ }={360}^{\circ }\\ \angle 5={360}^{\circ }-255^{\circ }\\ \angle 5=105^{\circ }\\ 105^{\circ }\ne {108}^{\circ }\end{array}$