Objectives (5 - 7 minutes)

1. Understand the concept of parallel lines cut by a transversal: Students should be able to understand the definition of these terms and how they relate in the context of geometry. This includes identifying parallel lines and a transversal, as well as understanding how the transversal cuts the parallel lines.

2. Identify and apply the main properties of parallel lines cut by a transversal: Students should be able to identify and use the main properties of these geometric figures. This includes applying the Theorem of Thales and identifying corresponding angles, alternate angles, and alternate interior angles.

3. Solve practical problems involving parallel lines cut by a transversal: Students should be able to apply the acquired knowledge to solve practical problems. This includes solving problems that involve calculating angles and identifying properties of geometric figures.

Secondary Objectives:

• Develop logical and analytical thinking skills: Through the study of geometry, students will have the opportunity to develop their logical and analytical thinking skills, which are valuable skills in many areas of life.

• Promote understanding of the importance of mathematics in everyday life: By applying the knowledge of geometry to practical problems, students will be able to understand the relevance of mathematics in daily life. This can help motivate them to engage more actively in learning the discipline.

Introduction (10 - 15 minutes)

1. Review of previous concepts (3 - 5 minutes): The teacher should start the lesson by reviewing the concepts of angles, parallelism, and basic geometry that were previously taught. This includes the definition of angles, types of angles (acute, right, obtuse), the sum of the angles of a triangle, parallel lines, and what it means for two lines to be parallel.

2. Problem situation 1 (3 - 5 minutes): The teacher can propose the following situation: "Imagine you are building a park and need to place two parallel paths that intersect with a third path. How would you ensure that the two main paths are parallel and how would you determine the angles where they intersect with the transversal?"

3. Contextualization (2 - 3 minutes): The teacher should emphasize that geometry and, specifically, the concept of parallel lines cut by a transversal, have practical applications in many fields, including architecture, civil engineering, interior design, and even computer games and 3D graphics.

4. Introduction to the topic (2 - 3 minutes): The teacher should then introduce the topic of the lesson - parallel lines cut by a transversal - explaining that, although it may seem like an abstract topic, it is fundamental to understanding many mathematical concepts and applications. The teacher may also mention that the topic is a natural extension of what students have already learned about parallelism and angles.

5. Curiosity (1 minute): To spark students' interest, the teacher can share two curiosities:

   • Curiosity 1: "Did you know that the concept of parallel lines cut by a transversal is so important that it has its own theorem, called the Theorem of Thales, which is used in many fields, including art? The Theorem of Thales is a fundamental tool in painting, photography, and even in creating visual effects in movies and computer games."
   • Curiosity 2: "Here's a challenge for you: Can you think of an example of parallel lines cut by a transversal in our daily lives? Think about it and we'll discuss it in the next part of the lesson."

The teacher should then move on to the next stage, where the concepts will be explored in more detail.

Development (20 - 25 minutes)

1. Theory Explanation (10 - 12 minutes):

   1.1. Definition of Parallel Lines and Transversals (2 - 3 minutes): The teacher should explain that parallel lines are two or more lines that never intersect, and that a transversal is a line that cuts two or more lines at different points.

   1.2. Theorem of Thales (3 - 4 minutes): The teacher should present the Theorem of Thales, which states that when a transversal cuts two parallel lines, the segments determined on the transversal are proportional. The teacher should illustrate this idea with graphic examples.

   1.3. Corresponding and Alternate Angles (2 - 3 minutes): The teacher should introduce the concepts of corresponding angles (which are equal) and alternate angles (which are on opposite sides of the transversal, but are not adjacent). The teacher should show examples of each type of angle.

   1.4. Alternate Interior Angles (2 - 3 minutes): The teacher should explain that alternate interior angles are pairs of angles that lie between the two parallel lines and in the same position relative to the transversal. The teacher should show examples of alternate interior angles and explain that they are always congruent (i.e., have the same value).

2. Exercise Resolution (10 - 13 minutes):

   2.1. Angle Identification Exercises (5 - 7 minutes): The teacher should present a series of figures showing parallel lines cut by a transversal and ask students to identify the different types of angles. Students should be encouraged to justify their answers using the theory learned.

   2.2. Angle Calculation Exercises (5 - 6 minutes): The teacher should then present a series of problems involving angle calculations. Students should be guided to use the Theorem of Thales and identify the different types of angles to solve the problems.

3. Discussion and Clarification of Doubts (5 - 7 minutes):

   3.1. Discussion of Exercises (3 - 4 minutes): The teacher should review the solutions to the exercises as a group, highlighting the strategies used by students to solve the problems and providing constructive feedback.

   3.2. Clarification of Doubts (2 - 3 minutes): The teacher should then open the class to questions and clarify any doubts students may have.

This is the moment to consolidate the acquired knowledge and ensure that students have a solid understanding of the topic.

Return (8 - 10 minutes)

1. Review of Concepts (3 - 4 minutes): The teacher should start the Return stage by reviewing the main concepts covered in the lesson. This includes the definition of parallel lines and transversals, the Theorem of Thales, and the different types of angles (corresponding, alternate, and alternate interior).

2. Connection to the Real World (2 - 3 minutes): The teacher should then highlight how the knowledge acquired in the lesson applies to the real world. This may include examples of everyday situations where parallel lines cut by a transversal can be observed, such as in street and road design, building architecture, or even in arranging furniture in a room.

3. Reflection on the Importance of the Subject (2 - 3 minutes): The teacher should propose that students reflect on the importance of the subject. Some questions that can be asked include:

   1. "Why is it important to understand the concept of parallel lines cut by a transversal?"
   2. "How can you apply this knowledge in your lives?"
   3. "What are the possible careers where this knowledge can be useful?"

4. Verification Questions (1 - 2 minutes): To assess students' understanding and promote reflection, the teacher can propose some verification questions. For example:

   1. "What are parallel lines and a transversal?"
   2. "How does the Theorem of Thales apply to parallel lines cut by a transversal?"
   3. "What are the different types of angles that can be formed when a transversal cuts parallel lines?"

5. Student Feedback (1 - 2 minutes): Finally, the teacher should request feedback from students about the lesson. This may include questions such as:

   1. "What did you find most interesting in today's lesson?"
   2. "What were the challenges you faced when trying to solve the exercises?"
   3. "Is there any question that you still don't fully understand?"

The Return is a crucial stage to consolidate learning and ensure that students have understood the key concepts. Additionally, by connecting the lesson content to the real world, students can see the relevance of what they have learned and feel more motivated to continue studying the subject.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should start the Conclusion by summarizing the main points covered during the lesson. This includes the definition of parallel lines and transversals, the Theorem of Thales, and the different types of angles formed when a transversal cuts parallel lines. The teacher should reinforce the most important concepts and clarify any remaining doubts.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher should then highlight the importance of connecting theory (mathematical concepts), practice (exercises solved during the lesson), and applications (how parallel lines and transversals are used in the real world). The teacher should emphasize that understanding the theory is fundamental to solving the exercises and that the ability to apply these concepts in practical situations is what makes mathematics relevant and useful.

3. Extra Materials (1 - 2 minutes): The teacher should suggest extra materials for students who wish to deepen their knowledge on the topic. This may include math books, educational websites, YouTube videos, and interactive online activities. The teacher should emphasize that self-study is an important part of the learning process and that these materials can help students consolidate what they have learned in the lesson.

4. Importance of the Topic for Everyday Life (1 minute): Finally, the teacher should reinforce the importance of the topic for everyday life. This may include examples of how the knowledge of parallel lines and transversals can be useful in everyday situations, such as solving design problems, understanding maps and diagrams, or even organizing furniture in a room. The teacher should end the lesson by reinforcing that mathematics is a practical and relevant discipline, and that understanding these concepts can be useful in many areas of life.