INTRODUCTION

Relevance of the Topic

Triangles are present in numerous aspects of the world around us, from architecture to nature. They are a basic geometric shape that is fundamental to understanding more complex concepts. Classifying triangles by angles is an essential skill for the study of Geometry. By classifying a triangle as acute-angled, right-angled, or obtuse-angled, we come to understand more than just its shape; we come to understand its properties, patterns, and behaviors.

Contextualization

The study of triangles is part of the Mathematics curriculum in High School, within the contents of Plane Geometry. This theme serves as a basis for further studies, such as triangle properties, trigonometry, Analytical Geometry, among others. Therefore, classifying triangles is a preliminary step towards building a solid understanding of Geometry. By familiarizing themselves with the names and properties of triangles according to their angles, students will gain tools to interpret and understand the world around them through shapes.

Remember: the triangle is, in fact, the simplest way to connect three points in space. Understanding its angular properties is a crucial step in the journey of unraveling the secrets that Geometry has to offer!

THEORETICAL DEVELOPMENT

• Components

  • Triangle: A geometric shape composed of three sides and three angles. It is the simplest shape that can be made with straight lines.

  • Angles: In the context of triangles, angles are the inclinations between the sides. Each triangle has exactly three angles, and the sum of all angles is always 180 degrees.

  • Classification by Angles: Triangles can be classified based on the measurement of their angles. Correctly identifying the angles is essential for this classification. There are three main categories: acute-angled, right-angled, and obtuse-angled.

• Key Terms

  • Acute-angled Triangle: A triangle where all angles are acute, meaning they are less than 90 degrees. The shape of this triangle can vary widely, from almost equilateral (all sides and angles equal) to very elongated.

  • Right-angled Triangle: This is a triangle that has a right angle, measuring exactly 90 degrees. The side opposite this angle is known as the hypotenuse, and it is always the longest side of the triangle.

  • Obtuse-angled Triangle: This is a triangle with an obtuse angle - an angle greater than 90 degrees and less than 180. This angle is larger than any angle possible in a right-angled or acute-angled triangle.

• Examples

  • Example of Acute-angled Triangle: A triangle with angles of 60 degrees. Note that all angle measurements are less than 90 degrees.

  • Example of Right-angled Triangle: The famous Pythagorean Triangle, with angles of 30, 60, and 90 degrees. The presence of the 90-degree angle classifies this triangle as right-angled.

  • Example of Obtuse-angled Triangle: A triangle with angles of 30, 50, and 100 degrees. The 100-degree angle is obtuse, classifying this triangle as obtuse-angled.

DETAILED SUMMARY

Key Points:

• Triangle Definition: It is crucial to understand that a triangle is a geometric shape composed of three sides and three angles, being the simplest shape that can be formed with straight lines.

• The Importance of Angles: Angles are the inclinations between the sides of a triangle. There will always be three angles in a triangle, and their sum will be 180 degrees.

• Classification of Triangles by Angles: Triangles can be classified by the measurement of their angles. This gives us three main categories: acute-angled, right-angled, and obtuse-angled.

  • Acute-angled Triangles: All angles are less than 90 degrees. Regardless of how elongated or balanced the triangle may appear, if all angles are less than 90 degrees, it is an acute-angled triangle.

  • Right-angled Triangles: Always have an angle of exactly 90 degrees and are easily identifiable by the presence of this characteristic. The side opposite the right angle is called the hypotenuse and is always the longest side of the triangle.

  • Obtuse-angled Triangles: These have an angle greater than 90 degrees and less than 180. In other words, it has an angle that is larger than any angle you would find in a right-angled or acute-angled triangle.

Conclusions:

• Classifying triangles by angles is a necessary tool for understanding geometric properties, behaviors, and patterns.

• Through angle classification, students can become familiar with the distinctive characteristics of right-angled, acute-angled, and obtuse-angled triangles and how to identify them.

Exercises:

1. Identify the type of triangle by its angles: 70º, 80º, and 30º.

2. A triangle has an angle of 120 degrees. What type of triangle is this?

3. Draw an example of each type of triangle, labeling the angles to demonstrate that you understand the classification.