Introduction

Relevance of the Topic

Polygon Angles is a fundamental concept in the study of Geometry. Polygons are everywhere in mathematics and real life. They are flat figures with straight sides. Understanding how angles are related to the sides of a polygon allows conclusions about the properties and characteristics of these figures. This understanding will be crucial for future studies of geometry, such as calculating areas and understanding other more advanced concepts, such as the congruence and similarity of polygons.

Contextualization

In our vast mathematical universe, polygon angles reside in the domain of plane geometry, a branch of mathematics that studies the properties, measurements, and relationships of two-dimensional figures, or what we know as drawings on a plane. This is a subsequent and crucial step in the study of angles in straight lines (180º), as we now delve into how these are used and related to more complex figures. Furthermore, the study of polygon angles lays the groundwork for the knowledge of other more advanced topics, such as trigonometry, calculation of areas and volumes, and relationships of congruence and similarity of figures.

Theoretical Development

Components

• Regular Polygons: These are polygons in which all sides and internal angles are congruent (equal). The characteristic of these figures simplifies the calculation of internal angles, since by dividing the total internal degrees by the number of sides, we obtain the exact value of each internal angle. Example: A square is a regular polygon, as all its sides and internal angles measure 90º.

• Formula for the Sum of Internal Angles: Expresses the relationship between the number of sides (n) of a polygon and the sum of its internal angles (Si). The formula is given by Si = (n-2) * 180º. This formula is useful for verifying and calculating angles in any polygon, regardless of whether it is regular or not.

• External Angle of a Polygon: It is the angle formed by one side of the polygon and the extension of the adjacent side. The sum of the external angles of any polygon, whether regular or not, is always 360º. Understanding this concept is vital to see the relationship between internal and external angles of a polygon.

Key Terms

• Polygon: It is a two-dimensional flat figure with straight sides, formed by at least three line segments that do not intersect. Polygons can be classified as regular or irregular, depending on whether their sides and internal angles are congruent or not.

• Internal Angle: It is the angle formed between two consecutive sides of a polygon. The sum of all internal angles of an n-sided polygon is given by the formula Si = (n-2) * 180º.

• External Angle: It is the angle formed by one side of the polygon and the extension of the adjacent side.

• Congruent (equal): It is a term used to describe angles or sides that have the same size or measurement.

Examples and Cases

• Square: This is an example of a regular polygon. All sides and angles of a square are congruent. Since it has four sides, the formula Si = (n-2) * 180º is used to calculate the sum of internal angles, resulting in 360º. If we divide 360º by the number of angles (4), we get the measure of each internal angle, which is 90º.

• Regular Hexagon: It has six sides and six angles, all of which are congruent. Therefore, the sum of the internal angles is: Si = (6-2) * 180º = 720º. Since it has 6 angles, each one measures 720º/6 = 120º. If we draw an external angle from each vertex of the hexagon, their sum will always be 360º.

• Irregular Pentagon: Although not a regular polygon, it still follows the formula Si = (n-2) * 180º. Thus, for a pentagon, its five internal angles sum Si = (5-2) * 180º = 540º. If the sides of the pentagon all have the same length, then each internal angle would be congruent and measure 540º/5 = 108º.

Detailed Summary

Key Points

• Definition of Polygons: Polygons are flat figures with straight sides and are among the most studied two-dimensional shapes. They are classified into two types: regular, where all sides and angles are congruent, and irregular, where this does not apply.

• Internal Angles of Regular Polygons: These are important because, in a regular polygon, all internal angles have the same measure. For example, in a square, all internal angles measure 90º.

• Sum of Internal Angles (Formula): All polygons, whether regular or irregular, have a formula that determines the sum of their internal angles. This formula is given by Si = (n-2) * 180º, where n is the number of sides of the polygon.

• External Angle of a Polygon: Knowledge about this angle is fundamental to establish the relationship between internal and external angles. Its sum, regardless of the type of polygon considered, will always be equal to 360º.

• Calculations and Relationships: Through the internal angles and the external angle of a polygon, we can perform various calculations, such as: determining the measure of any internal angle in a regular polygon, calculating the sum of internal angles in polygons in general, and verifying whether a polygon is regular or not.

Conclusions

• Understanding the internal and external angles of a polygon, together with the formula for the sum of internal angles, allows for various inferences about the properties and characteristics of polygons.

• In regular polygons, all internal angles have the same measure, which facilitates calculations and analyses of these figures.

• The study of polygon angles is a key step in the field of plane geometry, serving as a basis for other more advanced concepts and calculations, such as trigonometry, the area and perimeter of polygons, and the congruence and similarity of figures.

Proposed Exercises

1. Determine the measure of each internal angle of a regular pentagon.

2. Calculate the sum of the internal angles of a heptagon (a polygon with seven sides).

3. A polygon has 10 internal angles, each measuring 140º. How many sides does this polygon have?