Context

Triangles, geometric figures composed of three sides and three angles, might seem like a very basic concept at first, but they appear very frequently in the world around us and are very important in building mathematical foundations. In mathematics, every aspect of a triangle, be it its sides or its angles, holds a fascinating story.

In this project, we will focus on one such aspect, the sum of the interior angles of a triangle. Although it may seem simple, the idea that the interior angles of a triangle always add up to 180 degrees is a concept that has been crucial in many fields, ranging from the most basic arithmetic operations to very advanced concepts in geometry.

The discovery that the sum of the interior angles of a triangle is always 180° is one of the foundational principles of Euclidean geometry and has significant implications for many aspects of mathematics and physics. This simple concept enables the precise calculation of positions in three-dimensional space, the efficient design of physical structures, and the solution of complex problems in fields such as astrophysics and civil engineering.

In our surroundings, triangles are used frequently to provide stability in civil constructions, in navigation, where sailors use triangles to chart a course and locate themselves in relation to a reference point, and even in visual arts to create perspective and represent three-dimensional space on a two-dimensional surface.

To explore this topic further, we recommend the following resources:

• Triangles - Khan Academy
• "Geometry and Algebra" by Claudio Xavier. Scipione Publisher.
• Triangles - Sum of Angles - Brazil School

Hands-on Activity

Activity Title: The Sum of the Interior Angles of a Triangle

Project Goal: To understand and verify the mathematical property that states that the sum of the interior angles of a triangle is always 180 degrees.

Project Description

Students will work in groups of 3 to 5 and complete a series of hands-on activities to understand the sum of the interior angles of a triangle. Throughout the project, groups will make measurements, calculations, and engage in discussions to reach their conclusions. Students will work collaboratively to solve the proposed challenges, fostering collaboration and communication among group members.

Materials Required:

• Paper
• Pencils
• Protractor
• Scissors
• Ruler

Step-by-step

1. Each group should draw three different types of triangles (scalene, isosceles, and equilateral) on a piece of paper. The triangles should be large enough so that the angles can be measured using a protractor.

2. For each triangle, groups should measure the interior angles and record the values.

3. Groups should add up the interior angles of each triangle and record the results.

4. Groups should discuss why the sum of the interior angles of a triangle is always 180 degrees. They should research the theory behind this concept and include this discussion in their report.

Project Deliverables

The project should be submitted as a report that includes the following sections:

1. Introduction: Describe the purpose of the project and the importance of triangles and the angle property in mathematics and real life.

2. Development: Explain the theory behind the sum of the interior angles of a triangle. Then, describe in detail the step-by-step of the hands-on activity, the materials used, and the measurements and calculations performed. Also include discussions and comments made during the hands-on activity.

3. Conclusions: Summarize the main points of the report and write about what was learned from the project. Discuss how the hands-on activity helped to understand the angle property of a triangle.

4. Bibliography: Include all research sources used for the project. This can include books, websites, videos, etc.

This project aims to reinforce students' understanding of the sum of the interior angles of a triangle, in addition to developing skills such as teamwork, time management, problem-solving, and critical thinking.