Contextualization

Triangles are one of the most basic and fundamental structures in Geometry. They have a fascinating property: no matter the type of triangle, the sum of the interior angles is always 180 degrees. This is true for an equilateral triangle, an isosceles triangle, and even for a scalene triangle.

Introduction to the Study of Triangles

Triangle is one of the simplest yet one of the most important polygons in Mathematics. It is a closed plane figure made up of three straight line segments that intersect at three distinct points. The study of triangles and their angles are thus a basic topic in the development of students' geometric reasoning.

Within the study of triangles, there is the important issue of the sum of the interior angles. A profound understanding of this topic is crucial to advance in more complex concepts such as trigonometry, in spherical geometry and even in the study of geometric shapes in higher dimensions.

The Importance of Triangles in the Real World

The application of triangles and their angles is not restricted to mathematical theory only. It extends to several areas of practical life. For example, triangles and their properties are used by engineers and architects to design stable and durable structures. In navigation, triangles are used to determine the shortest path between two points on the Earth's surface, using spherical geometry.

Moreover, the issue of a triangle's interior angles also has implications in more advanced areas such as the Theory of Relativity. In Einstein's spacetime geometry, the universe is curved, and the sum of the angles of a triangle is greater than 180 degrees. This difference, although very small, has huge implications in the way we understand the universe.

Hands-on Activity: "Exploring Triangle Angles"

Project Objective

The goal of this project is to allow students to discover, for themselves, the sum of the interior angles of a triangle. They should be able to verify and realize that no matter the type, the sum of the interior angles of any triangle is always 180 degrees. Moreover, at the end of the project, students should be able to calculate the third angle of a triangle knowing the other two.

Detailed Description of the Project

This project will be carried out by groups of 3 to 5 students and will take a month. During this time, students will build different types of triangles, measure their angles, and verify the 180-degree sum property.

Required Materials

In order to carry out this project, students will need the following materials:

• Ruler
• Compass
• Graph paper
• Pencil
• Protractor

Detailed Step-by-Step

1. Divide the class into groups of 3 to 5 students.
2. Each group should draw, on paper, at least three different types of triangles (equilateral, isosceles, scalene) using the compass and the ruler.
3. Students should measure the interior angles of each triangle using the protractor.
4. They should add up the interior angles of each triangle and write down the result.
5. Each group should record the measurements of all angles of all the triangles they drew, in a table.
6. Students should reflect on the results obtained and try to reach a conclusion.
7. Finally, students should write a report on the activities carried out, describing step by step the whole process, presenting the results obtained and their conclusions.

Project Deliverables and Written Report

At the end of the project, each group must deliver:

1. The drawings of the triangles with the interior angle measurements duly annotated.
2. The table containing the angle measurements of all the triangles drawn by the group.
3. The written report on the project. This document should contain:
   • Introduction: Students should contextualize the topic, its relevance, and application in the real world, as well as the objective of this project.
   • Development: Students should make explicit the theory behind the project's central theme, explain the activity in detail, indicate the methodology used, and finally present and discuss the results obtained.
   • Conclusion: Students should conclude the work by resuming their main points, making explicit the learning achieved, and the conclusions drawn about the project.
   • Bibliography: Students should indicate the sources on which they based themselves to work on the project such as books, web pages, videos, etc.

This report should be delivered in digital format and, in case of citation of any third-party resource, it is important that students make proper reference to the source, following the ABNT standards.