Chapter PS, Problem 1.3P

To determine

To find:

1. To determine the sum of angles of a quadrilateral, pentagon and hexagon using the sum of angles of a triangle.
2. Rule to find sum of the measures of the angles of an n-gon.

Answer to Problem 1.3P

 a. $360°$ , $540°$ , $720°$

 b. $180\times \left(n-2\right)°$

Explanation of Solution

Given:

Sum of all the angles of a triangle is equal to $180°$ .

  

Referring to above quadrilateral, there are two triangles.

Using angle sum property of both the triangles, it can be written that:

  $\angle 2+\angle B+\angle 3=180°$

  $\angle 1+\angle D+\angle 4=180°$

Adding both the equations:

  $\angle A+\angle B+\angle C+\angle D=360°$

  

Referring to above Pentagon, there are three triangles.

Using angle sum property in all three triangles, it can be written that:

  $\angle 1+\angle B+\angle 2=180°$

  $\angle 3+\angle 4+\angle 5=180°$

  $\angle E+\angle 6+\angle 7=180°$

Adding all three equations:

  $\angle A+\angle B+\angle C+\angle D+\angle E=540°$

  

Referring to above Hexagon, there are four triangles.

Using angle sum property in all three triangles, it can be written that:

  $\angle 1+\angle B+\angle 2=180°$

  $\angle 3+\angle 4+\angle 5=180°$

  $\angle 6+\angle 7+\angle 8=180°$

  $\angle F+\angle 9+\angle 10=180°$

Adding all four equations:

  $\angle A+\angle B+\angle C+\angle D+\angle E+\angle F=720°$

It can be seen that when 180 $°$ is multiplied by the (number of sides of the polygon minus 2), the internal angle sum of the n-gon is obtained.

So, the rule is:

Internal angle sum of an n-gon= $180\times \left(n-2\right)°$