Chapter 10.1, Problem 27E

To determine

To find the sum of the measures of the angles of the regular hexagon.

Answer to Problem 27E

The sum of the measure of the angles of the hexagon is ${720}^{\circ }$

Explanation of Solution

Given information:

The triangle is an equilateral triangle and need to create 6 equal triangle to form hexagon.

Calculation:

A regular hexagon created as below by placing 6 copies of an equilateral triangle.

Sum of measures of the angles of the hexagon can be written as :

  $\angle AFE+\angle FED+\angle EDC+\angle DCB+\angle CBA+\angle BAF$

Each of the above angles is made up of two angles of the equilateral triangles, as mentioned below

  $\angle AFE=\angle AFO\text{}+\text{}\angle OFE$

Since $\angle AFO$

  $={60}^{\circ }$ and $\angle OFE={60}^{\circ }$

  ${60}^{\circ }+{60}^{\circ }={120}^{\circ }$

  $\angle FED=\angle FEO\text{}+\text{}\angle OED$

Since $\angle FEO\text{}$

  $={60}^{\circ }$ and $\angle OED={60}^{\circ }$

  ${60}^{\circ }+{60}^{\circ }={120}^{\circ }$

  $\angle EDC=\angle EDO\text{}+\text{}\angle ODC$

Since $\angle EDO\text{}$

  $={60}^{\circ }$ and $\angle ODC={60}^{\circ }$

  ${60}^{\circ }+{60}^{\circ }={120}^{\circ }$

  $\angle DCB=\angle DCO\text{}+\text{}\angle OCB$

Since $\angle DCO$

  $={60}^{\circ }$ and $\angle OCB={60}^{\circ }$

  ${60}^{\circ }+{60}^{\circ }={120}^{\circ }$

  $\angle CBA=\angle CBO\text{}+\text{}\angle OBA$

Since $\angle CBO$

  $={60}^{\circ }$ and $\angle OBA={60}^{\circ }$

  ${60}^{\circ }+{60}^{\circ }={120}^{\circ }$

  $\angle BAF=\angle BAO\text{}+\text{}\angle OAF$

Since $\angle BAO$

  $={60}^{\circ }$ and $\angle OAF={60}^{\circ }$

  ${60}^{\circ }+{60}^{\circ }={120}^{\circ }$

As can be seen above, each of the 6 angles of the regular hexagon measures ${120}^{\circ }$

Hence sum of the measures of the angles of the regular hexagon is

  $\angle AFE+\angle FED+\angle EDC+\angle DCB+\angle CBA+\angle BAF$

Now , replace each of the angles by ${120}^{\circ }$

  ${120}^{\circ }+{120}^{\circ }+{120}^{\circ }+{120}^{\circ }+{120}^{\circ }+{120}^{\circ }$

By adding all the terms above, get the final answer as ${720}^{\circ }$

Conclusion- The sum of the measures of the angles of the regular hexagon is ${720}^{\circ }$