Problem 33 - Algebra and Trigonometry

Question

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 \((n-2) \cdot 180^{\circ} .\)

Step-by-Step Solution

Answer

The sum of the interior angles of a convex polygon with n sides is \((n-2) \times 180^{\circ} \).

Step 1: Base Case

Consider the base case where the convex polygon has 3 sides, i.e., a triangle. A triangle has the sum of its interior angles as \( (3-2) \times 180^{\circ} = 1 \times 180^{\circ} = 180^{\circ} \). Hence, the statement holds true for a triangle.

Step 2: Inductive Hypothesis

Assume that the formula holds for a convex polygon with k sides. This means the sum of the interior angles for a polygon with k sides is \((k-2) \times 180^{\circ} \).

Step 3: Inductive Step

Now, consider a polygon with (k+1) sides. By adding one vertex to the polygon with k sides, a triangle is formed within the polygon. The sum of the interior angles of the (k+1)-sided polygon is the sum of the interior angles of the k-sided polygon plus 180° (the interior angles of the added triangle). This gives: \((k-2) \times 180^{\circ} + 180^{\circ} = (k-1) \times 180^{\circ}\) Hence, the formula holds for (k+1) sides.

Step 4: Conclusion

By the Extended Principle of Mathematical Induction, since the formula holds for a polygon with 3 sides and for (k+1) sides assuming it holds for k sides, the formula is true for any convex polygon with n sides. Thus, the sum of the interior angles of a convex polygon with n sides is \((n-2) \times 180^{\circ} \).

Key Concepts

Extended Principle of Mathematical Induction

The principle of mathematical induction is a powerful technique used to prove statements for all natural numbers. Generally, it involves two steps: proving the base case and the inductive step. However, the Extended Principle of Mathematical Induction (EPMI) extends this concept further, allowing the assumption to carry over more comprehensively.

To use EPMI:

• First, establish that the statement holds for the initial case, often called the base case. In our exercise, this would be a triangle (a polygon with 3 sides).


• Next, form the inductive hypothesis by assuming the statement is true for a polygon with k sides.


• Then, use this assumption to prove the formula is also true for a polygon with (k+1) sides. By establishing these steps, you demonstrate the truth of the statement for all natural numbers greater than or equal to 3 (or any starting number in different contexts).

In our exercise, the EPMI helps show the formula for the sum of interior angles holds for any convex polygon, which forms the cornerstone for further applications in geometry.

Convex Polygon

A convex polygon is defined as a polygon where all its interior angles are less than 180 degrees, and none of the vertices point inward. This makes it easier to apply formulas and principles since the shape is uniform.

Key characteristics of convex polygons include:

• Each interior angle is less than 180°.


• The polygon doesn’t have any dented-in edges.


• The line segment between any two points within the polygon remains inside the polygon.

Understanding these characteristics helps in applying various geometric principles accurately. For instance, when adding a vertex to a convex polygon and forming a new side, each new side connects to form additional triangles.

Geometry

Geometry is the branch of mathematics concerned with shapes, sizes, relative positions, and properties of space. To solve the problem using geometry, knowing the properties of polygons is crucial.

Some essential geometry principles include:

• The sum of the interior angles of a polygon is derived by dividing the polygon into triangles. Each triangle has a sum of interior angles equal to 180°.


• The formula for the interior angles of an n-sided polygon is found by recognizing that an n-sided polygon can be triangulated into (n-2) triangles.


• For our problem, this property is utilized by showing that stepping from a k-sided polygon to a (k+1)-sided polygon involves adding a triangle, which introduces an additional 180° to the sum of angles.
  
  The application of these geometric properties and principles makes solving such problems systematic and reinforces understanding through visual and conceptual clarity.