Question

Use the Extended Principle of Mathematical Induction to show that the sum of the interior angles of a convex polygon of n sides equals$\left(n-2\right)180°$

Short answer

The given statement is proved.

Step-by-step solution

Step 1. Given information

The Extended Principle of Mathematical Induction is to be used.

Step 2. Prove the given statement.

For a polygon, the least number of sides is 3 and the sum of the angles is $180°$

Put $n=3$in $(n-2)·180°$to check if the statement is true for width="37">n=3

width="156">(3-2)·180°=1·180°=180°

Thus, the statement is true for $n=3$

Assume that the statement is true for n = k.

Thus, the sum of the angles of a convex polygon with k sides is $(k-2)·180°$.

Find the sum of the angles of a convex polygon with k + 1 sides and check whether the sum equals $\left(\right(k+1)-2)·180°$

A convex polygon with k + 1 sides consists of a convex polygon with k sides and also a triangle.

Thus the sum of the angles is

localid="1647850422292" $\begin{array}{rcl}(k-2)·180°+180°& =& (k-2+1)·180°\\ & =& \left[\right(k+1)-2]·180°\end{array}$

Thus, the statement is true for k + 1.

Since both the conditions of the Principle of Mathematical Induction are satisfied, the statement is true for all natural numbers.

Therefore, the sum of the interior angles of a convex polygon of n sides equals$(n-2)·180°$