Lesson Plan | Traditional Methodology | Triangles: Congruence

Objectives

Duration: (10 - 15 minutes)

The purpose of this lesson plan stage is to ensure that students clearly understand the fundamental concepts of triangle congruence, preparing them to deepen their knowledge and apply these concepts in problem-solving. This solid foundation will allow them to follow along and actively participate in the subsequent explanations and activities.

Main Objectives

1. Understand that two congruent triangles are those that have equal angles and sides.

2. Know the main cases of triangle congruence (SSS, SAS, ASA, AAS) and apply them to solve problems.

Introduction

Duration: (10 - 15 minutes)

The purpose of this lesson plan stage is to motivate students, awakening their interest and curiosity about the topic. By presenting the context and curiosities, students can realize the relevance of studying congruent triangles and feel more engaged to learn the concepts that will be addressed throughout the lesson.

Context

To start the class on triangle congruence, begin by highlighting the importance of triangles in mathematics and in our daily lives. Explain that triangles are the simplest geometric shape that can be used to divide other more complex shapes, being widely used in areas such as architecture, civil engineering, and design. Mention that understanding triangles and their properties is fundamental to solving practical and theoretical problems in various disciplines.

Curiosities

Did you know that the ancient Egyptians used congruent triangles to build the pyramids? They used ropes with knots at equal intervals to create perfect angles and ensure that all faces of the pyramids had the same measurements, which demonstrates the practical use of triangle congruence in engineering and construction.

Development

Duration: (60 - 70 minutes)

The purpose of this lesson plan stage is to ensure that students thoroughly understand the cases of triangle congruence and know how to apply them in problem solving. By exploring each case with examples and geometric proofs, and solving guided problems, students will develop a solid practical and theoretical understanding, essential for mastering the topic.

Covered Topics

1. Definition of Congruent Triangles: Explain that two triangles are congruent when they have equal corresponding angles and sides. Use visual examples to illustrate congruence. 2. Cases of Triangle Congruence: Detail the four main cases of triangle congruence: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS). Provide examples and geometric proofs for each case: 3. Side-Side-Side (SSS): Three sides of one triangle are equal to the three sides of another triangle. 4. Side-Angle-Side (SAS): Two sides and the angle between them in one triangle are equal to those in another triangle. 5. Angle-Side-Angle (ASA): Two angles and the side between them in one triangle are equal to those in another triangle. 6. Angle-Angle-Side (AAS): Two angles and a non-included side of one triangle are equal to those in another triangle. 7. Practical Applications: Discuss how triangle congruence is used in real problems, such as in civil construction, architecture, and engineering. Provide examples of practical situations where triangle congruence is applied. 8. Guided Problem Solving: Present problems for resolution in class, explaining step by step the application of the cases of congruence to solve these problems. Include problems that involve different cases of congruence and encourage students to follow along with the resolution.

Classroom Questions

1. 1. Two triangles have corresponding sides equal to 6 cm, 8 cm, and 10 cm. Are they congruent? Justify your answer. 2. 2. If two triangles have two corresponding angles equal and the side between those angles is also equal, what case of congruence does this represent? Solve a numerical example. 3. 3. Given a triangle with sides of 5 cm, 12 cm, and 13 cm, determine if it is congruent to another triangle that has angles of 30°, 60°, and 90° and one of its sides measuring 5 cm. Explain your reasoning.

Questions Discussion

Duration: (15 - 20 minutes)

The purpose of this lesson plan stage is to review and consolidate students' understanding of triangle congruence concepts. Discussing the answers to the questions allows students to clarify doubts, validate their reasoning, and reinforce learning through collective engagement and teacher feedback.

Discussion

• 1. Two triangles have corresponding sides equal to 6 cm, 8 cm, and 10 cm. Are they congruent? Justify your answer. Explanation: Yes, the triangles are congruent. By the SSS condition (Side-Side-Side), we know that if all three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent. In this case, since the corresponding sides are equal (6 cm, 8 cm, and 10 cm), the triangles are congruent.
• 2. If two triangles have two corresponding angles equal and the side between those angles is also equal, what case of congruence does this represent? Solve a numerical example. Explanation: This represents the ASA case (Angle-Side-Angle). Example: Consider two triangles where the angles are 45° and 60°, and the side between those angles is 7 cm. If the angles and the side between them are equal in both triangles, the triangles are congruent by the ASA case.
• 3. Given a triangle with sides of 5 cm, 12 cm, and 13 cm, determine if it is congruent to another triangle that has angles of 30°, 60°, and 90° and one of its sides measuring 5 cm. Explain your reasoning. Explanation: No, the triangles are not congruent. The first triangle has sides of 5 cm, 12 cm, and 13 cm, while the second triangle has angles of 30°, 60°, and 90° and one side of 5 cm. For them to be congruent, the triangles would need to have all corresponding sides and angles equal. In this case, the angles and sides do not consistently match with any of the cases of congruence (SSS, SAS, ASA, AAS).

Student Engagement

1. 1. Why is the SSS condition sufficient to guarantee the congruence of triangles? Can you think of a practical example? 2. 2. If two triangles have two equal angles and a non-included side equal, why is this not sufficient to guarantee congruence? What case of congruence would be necessary? 3. 3. How can we use triangle congruence to solve real-life problems, such as in civil construction or architecture? Can anyone give an example?

Conclusion

Duration: (10 - 15 minutes)

The purpose of this lesson plan stage is to review and consolidate students' learning by recapping the main points covered during the lesson and reinforcing the importance of the topic. In addition, this stage aims to connect theory with practical applications, demonstrating the relevance of the concept of triangle congruence in real-world contexts and various fields of knowledge.

Summary

• Two triangles are congruent when they have equal corresponding angles and sides.
• The main cases of triangle congruence are: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS).
• Congruence is used in various practical areas such as civil construction, architecture, and engineering.
• Problem-solving involving triangle congruence with practical examples for each case.

The lesson connected theory with practice by detailing the cases of triangle congruence and demonstrating how these concepts are applied in real situations, such as constructing structures and solving geometric problems, facilitating students' understanding of the importance and utility of triangle congruence in the real world.

The study of triangle congruence is crucial in daily life as it allows for solving problems in various fields such as engineering, architecture, and design. For example, when designing a building, ensuring that structural parts are congruent is essential for stability and safety. Furthermore, understanding this concept can help students develop analytical and problem-solving skills.