Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-42: Prove that the statement is true for every positive integer \(n\). Prove that for every positive integer \(n \geq 3\), the sum of the interior angles of an \(n\)-sided polygon is given by the expression \((n-2) \cdot 180^{\circ}\).

Step-by-Step Solution

Answer

The sum of the interior angles of an \(n\)-sided polygon is \((n-2) \cdot 180^\circ\) by dividing the polygon into \((n-2)\) triangles.

Step 1: Understanding the Problem

We need to prove that for any polygon with \(n\) sides (where \(n\) is a positive integer greater than or equal to 3), the sum of its interior angles can be calculated using the formula \((n-2) \cdot 180^\circ\).

Step 2: Base Case Verification

Let's verify the formula for an \(n=3\) sided polygon, which is a triangle. The sum of interior angles of a triangle is known to be \(180^\circ\). Plugging \(n = 3\) into the formula \((n-2) \cdot 180^\circ = (3-2) \cdot 180^\circ = 180^\circ\), which confirms our observation.

Step 3: Understanding Interior Angle Sum

Consider an \(n\)-sided polygon. By dividing it into \((n-2)\) triangles (by drawing diagonals from one vertex to all other non-adjacent vertices), the sum of the interior angles of these triangles equals the sum of the interior angles of the polygon.

Step 4: Deriving the Formula

Since each triangle has a sum of interior angles of \(180^\circ\), the sum for \((n-2)\) triangles is \((n-2) \times 180^\circ\). This verifies our formula for any polygon with \(n\) sides.

Key Concepts

Positive Integer

A positive integer is simply a whole number greater than zero. This means numbers like 1, 2, 3, and so on. In the context of polygons, especially when discussing sides, the number of sides a polygon has is denoted by a positive integer.

• For instance, a triangle has 3 sides, making it a 3-sided polygon, satisfying the condition that it involves positive integers.
• Similarly, a rectangle, having 4 sides, represents a positive integer as well.

Understanding positive integers is vital for geometric proofs involving polygons, as it ensures clarity when dealing with the quantitative aspects of these shapes. It sets the stage for deriving and proving formulas that relate to the properties of these geometric figures.

Interior Angles Sum

The interior angles sum of a polygon relates to the sum of angles within the polygon. For polygons with different numbers of sides, this sum can be determined by using the formula: \((n-2) \cdot 180^\circ\),where \(n\) represents the number of sides, which must be a positive integer.

• This formula arises from dividing the polygon into triangles from a chosen vertex. Each triangle within the polygon contributes a sum of \(180^\circ\) to the total interior angle sum.
• For instance, a quadrilateral can be divided to form two triangles, resulting in an interior angle sum calculated as \((4-2) \cdot 180^\circ = 360^\circ\).

Understanding the interior angles sum allows students to see relationships between shapes and enhances problem-solving skills in geometry involving polygons.

Geometric Proof

A geometric proof is a step-by-step reasoning process used to demonstrate the truth of a mathematical statement, often utilizing basic geometric concepts, definitions, and axioms. In this context, the task is to prove the formula \((n-2) \cdot 180^\circ\) for any \(n\)-sided polygon using geometric reasoning.

• First, verify the base case: For a triangle \((n=3)\), the interior angles sum to \(180^\circ\), aligning with the formula.
• Then, progress with a reasoning step: By dividing a polygon into \((n-2)\) triangles, each contributing \(180^\circ\), validates the formula for all \(n\) greater than 3.

The strategy of breaking down complex shapes (polygons) into familiar/usual shapes (triangles) leverages known truths to establish broader geometric principles. This not only proves the rule but also deepens understanding and fosters logical reasoning.