Chapter 12.4, Problem 33AYU

Geometry Use the Extended Principle of Mathematical Induction to show that the sum of the interior angles of a convex polygon of $n$ sides equals $\left(n-2\right)\cdot {\text{ 180}}^{\circ }$.

To determine

To prove: The statement $c$ is true for all natural numbers $n$ using the Extended Principle of Mathematical Induction.

Answer to Problem 33AYU

The statement is true for the natural number $\left($ k + 1$\right)$ terms, it is true for all natural numbers $\ge j$ by the extended principle of mathematical induction.

Explanation of Solution

Given:

Statements says ‘The sumof the interior angles of a convex polygon of $n$ sides is $\left(n - 2\right).180°$’ is true for all natural number.

Formula used:

The Extended Principle of Mathematical Induction

Suppose that the following two conditions are satisfied with regard to a statement about natural numbers:

CONDITION I: A statement is true for a natural number $j$.

CONDITION II: If the statement is true for some natural number $k \ge j$, it is also true for the next natural number $k + 1$. Then the statement is true for all natural numbers $\ge j$.

Proof:

Consider the statement

‘The sumof the interior angles of a convex polygon of $n$ sides is $\left(n - 2\right).180°$’   -----(1)

Step 1: Show that statement (1) is true for the natural number $j = 3$

Number of sides = 3. 3-sided polygon is nothing but a triangle.

The sum of the angles of a triangle is $180°$. The sum of the interior angles of a convex polygon of 3 sides is $\left(3 - 2\right)180 = 180°$ Hence the statement is true for the natural number $j = 3$.

Step 2: Assume that the statement is true for some natural number $k$.

That is ‘The sumof the interior angles of a convex polygon of $k$ sides is $\left(k - 2\right).180°$’    -----(2)

Step 3: Prove that the statement is true for the next natural number $k + 1$.

‘The sumof the interior angles of a convex polygon of $n$ sides is $\left(k + 1 - 2\right).180° = \left(k - 1\right)180°$’

From equation (2), it can be concluded that for $k$ sides, the sum of the interior angles of a convex polygon is $\left(k - 2\right).180°$.

A convex polygon of $k + 1$ sides consists of a convex polygon of $k$ sides plus a triangle (see the illustration).

‘The sumof the interior angles of a convex polygon of $n$ sides is $\left(k - 2\right).180° + 180° = \left(k - 2 + 1\right)180° = \left(k - 1\right)180°$

As the statement is true for the natural number $\left($ k + 1$\right)$ terms, it is true for all natural numbers $\ge j$ by the extended principle of mathematical induction.