Chapter 9.4, Problem 76E

To determine

Use mathematical induction to prove this formula for a general $n$ -sided polygon.

Answer to Problem 76E

  $\mathrm{Pr}oved$

Explanation of Solution

Calculation:

Consider the given figures.

  

Consider triangle,

Sum of the angles, $180°=\left(3-2\right)180°$

Now consider square,

Sum of the angles $360°=\left(4-2\right)180°$

Sum of the angles of polygon,

  $540°=\left(5-2\right)180°$

So that the formula for sum of the angle in regular polygon of $n$ sides are,

  $\left(n-2\right)180°$

Apply mathematical induction,

Let $S_n:$ Sum of angles in a regular polygon of $n$ sides is $\left(n-2\right)180°$

First prove the statement for triangle $n=3$

  $\left(3-2\right)180°=180°$

Hence, formula is valid for triangle.

Now assume that formula is valid for some integer $k$ that is,

  $S_k:$ Sum of angles in a regular polygon of $k$ sides is $\left(k-2\right)180°$

Now let see $S_{k+1}$ is a valid statement,

That is you must prove sum of angles in a regular polygon of $\left(k+1\right)$ sides is $\left(k-1\right)180°$

Consider the polygon of $\left(k+1\right)$ sides

  

The polygon is divided in two patrs by diagonal as shown, in triangle

By induction method sum of angles in a polygon of $k$ sides is $\left(k-2\right)180°$

And sum of angles in a triangle is $180°$

So total sum is

  $\left(k-2\right)180°+180°=\left(k-1\right)180°$

Hence, $\mathrm{Pr}oved$ .