Contextualization

The congruence of triangles is a fundamental topic in Euclidean geometry, a branch of mathematics that deals with the shape, size, and relative position of figures and spaces. This theory, proposed by Euclid, a Greek mathematician from the 3rd century BC, formed the basis for much of our current understanding of mathematics and geometry.

Basically, the congruence of triangles states that two triangles are congruent if and only if their corresponding elements (sides and angles) are equal. There are several criteria for the congruence of triangles, according to the relationship of their elements, such as: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), among others. These concepts require a solid and in-depth understanding of the discipline, which is essential for delving into other more complex aspects of geometry.

The study of triangle congruence is not limited to the theoretical field but finds various practical applications in the real world, spanning various sectors such as civil engineering, architecture, maritime and aeronautical navigation, etc. For example, in civil engineering and architecture, triangle congruence is often used in planning and construction to ensure that structures are solid, stable, and symmetrical. In aerial and maritime navigation, triangle congruence is used to calculate distances and projections.

Furthermore, triangle congruence and geometry as a whole are a fundamental part of logical-mathematical thinking, which helps us better understand the world around us and develop essential skills such as logical reasoning, problem-solving, and critical thinking. Understanding triangle congruence, therefore, not only helps improve your mathematical skills but also develops essential skills that will be useful in many aspects of life.

For a theoretical deepening on the subject, I recommend reading the book "Euclidean Plane Geometry" by José Carlos Ciorlin and visiting the website Just Mathematics (justmathematics.com). Additionally, for a more practical and applied view of triangle congruence, the Education World portal (educationworld.com) brings examples and solved problems on the subject.

Practical Activity: Designing with Congruent Triangles

Project Objectives:

I. Understand and apply the concepts of triangle congruence in a practical context. II. Develop problem-solving, critical and creative thinking skills. III. Enhance socio-emotional skills such as teamwork, communication, and time management.

Project Description:

In this project, you will be divided into groups of 3 to 5 students and will have to create an architectural model of a building using congruent triangles. This building can be a house, a skyscraper, a church, a stadium, etc. The model is entirely up to the group's choice.

The project will be divided into two stages:

1. Theoretical Project: Sketch the architectural model on a grid paper indicating the measurements of the triangles that will be used and justifying their choice. Indicate which congruence criterion was used for the building's design.

2. Practical Project: Construction of the 3D model of the building using cardboard, colored paper, glue, and other available materials.

Required Materials:

• Grid paper
• Ruler, protractor, and compass
• Cardboard and colored paper
• Scissors and glue

Project Step by Step:

1. Choose the type of building to be designed and sketch a draft on a piece of paper.

2. Draw the architectural plan of the building on grid paper, making sure to use congruent triangles in its structure. Indicate the measurements of the sides and angles of the triangles used.

3. Justify the choice of triangles used and indicate which congruence criterion was used for the building's design.

4. After the theoretical project is approved, start building the 3D model of the building using cardboard, colored paper, and glue. Make sure to follow the measurements indicated in the theoretical project.

5. Upon completing the construction, make an oral presentation to the class exposing the project and explaining its construction process and the use of triangle congruence concepts.

Project Delivery:

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

1. The original theoretical project, complete with sketches and justifications.

2. The 3D architectural model.

3. A written document containing:

   a. **Introduction**: Description of the chosen building, justifying the choice and the application of triangle congruence in the project's design.
   
   b. **Development**: Detailing of the theory used in the project's design, description of the model's construction, the methodology used by the group, and discussion of the results obtained.
   
   c. **Conclusion**: Final reflections on the project, lessons learned, and application of triangle congruence concepts in real life.
   
   d. **Bibliography**: Reference to all consultation materials used for the project's design.