Learning Geometry with the GeoGebra Platform

Jorge Cássio, Aug 30, 2019

"Learning Geometry with the GeoGebra Platform" is a dynamic math book that explores many of Geometry's subjects. The book is designed to be explored with undergraduate students in mathematics, but it can also be used with students at the basic level.

Table of Contents

  1. About the Book and Author
    1. Jorge Cássio
    2. Acknowledgements
    3. The book (not translated)
    4. Teacher Guidelines (not translated)
    5. Student Guidelines (not translated)
  2. Basic Notions
    1. Primitive Notions and Propositions
    2. Position of point and straight line
    3. Segments and Midpoint
    4. Rays, angles and angle bisector
    5. Video lesson on Angles and Angle Bisector.
    6. Chapter 2 exercises
  3. Triangles
    1. Triangles: Definition and Elements
    2. Triangles- Classification
    3. Triangular Inequality – building the triangle
    4. Video classes about triangles
    5. CHAPTER 3 EXERCISES
  4. Congruence of Triangles
    1. Congruence of Triangles
    2. Triangle Congruence Cases
    3. Why does the ASS (or SSA) can't be used to determine triangle congruence?
    4. Solved Exercises
    5. Video lesson on congruence of triangles
    6. Chapter 4 exercises
  5. Parallelism and Perpendicularity
    1. Parallelism
    2. Perpendicularity
    3. Video lessons on parallelism and perpendicularity
    4. Chapter 5 exercises
  6. The Classical Triangle Centers
    1. Incenter
    2. Circumcenter
    3. The Orthocenter
    4. Barycenter
    5. The Euler line and the center of mass of the triangle (centroid)
    6. Video lessons on notable points
    7. Chapter 6 exercises
  7. Quadrilaterals - Definitions and Properties
    1. Quadrilateral-Basic Notions
    2. Trapezoid
    3. Parallelogram
    4. Rectangle
    5. Rhombus
    6. Square
    7. Triangle Midsegment
    8. Trapezoid Midsegment
    9. Video Lessons about Quadrilaterals (put English subtitles)
    10. Chapter 7 - Exercises
  8. Thales' Theorem and Angle Bisectors' Theorem
    1. Thales' Theorem
    2. Interior Angle Bisector Theorem
    3. Exterior angle bisector
    4. Dynamic Solved Exercise
    5. Video lesson on the Thales theorem
    6. Exercises from Chapter 8
  9. Circumference, Circle and Angles in the Circumference
    1. Circumference and Circle
    2. Congruent Arcs and Arcs Addition
    3. Central Angle, Inscribed Angle, Circular Segment Angle and Arc
    4. Inscribed quadrilateral and eccentric angles
    5. Video Lessons - Definitions, Relative Positions, etc.
    6. Chapter 9 Exercises
  10. Triangle Similarity, Power of a point, and Metric Relations in the Right Triangle
    1. Triangle similarity: Definition, criteria and demonstrations
    2. Power of a point
    3. Right-Triangle Elements, Metric Relations and Pythagorean Theorem
    4. Videoaulas sobre Semelhança de triângulos, Relações métricas no triângulo retângulo e Potência de Ponto
    5. Exercícios do capítulo 10
  11. Area of Rectangle, Square, Parallelogram, Trapezoid, Rhombus, Triangles and Regular Polygons.
    1. Rectangle Area
    2. Square Area
    3. Parallelogram Area
    4. Triangle Area
    5. Trapezoid Area
    6. Rhombus Area
    7. Regular Polygon Area
    8. Exercícios do Capítulo 11
  12. Circle
    1. Length of the Circumference
    2. Area of a Circle
    3. Video lesson: Circumference and Circle
    4. Chapter 12 exercises
  13. Geometric Drawing
    1. Point, Point of Intersection and Midpoint
    2. Tracing Angles and Angles in Radians
    3. Parallel, Perpendicular, Line Bisector and Angle Bisector
    4. Regular Polygons
    5. Videoaulas sobre polígonos regulares
    6. Exercícios do capítulo 13
  14. References
    1. Bibliographic References and Sites

1.

About the Book and Author

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

  • 1. Jorge Cássio
  • 2. Acknowledgements
  • 3. The book (not translated)
  • 4. Teacher Guidelines (not translated)
  • 5. Student Guidelines (not translated)

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2.

Basic Notions

The purpose of this chapter is to provide conditions for students to develop the following skills: • Understanding of the primitive notions and propositions. • Understanding of the definitions and properties related to segment, ray, midpoint, half-plane, angle and angle bisector. In order to do so, we will be exploring activities that allow the improvement of the following skills: • Knowing what is a proposition, a postulate, an axiom and a theorem. • Understanding of the postulates of existence, determination and inclusion. • Knowing and understanding how to associate the different representations of point, line and plane. • Knowing what is a line segment and recognizing its different representations. • Knowing how to differentiate consecutive segments, collinear segments and adjacent segments. • Knowing and understanding how to dupplicate line segments. • Knowing what is a ray, knowing its different representations. • Knowing what is the midpoint of a segment and how to determine it, when using a ruler and compass. • Knowing what is a half-plane and recognizing its different representations. • Knowing what is an angle is and measuring it using a protractor.

  • 1. Primitive Notions and Propositions
  • 2. Position of point and straight line
  • 3. Segments and Midpoint
  • 4. Rays, angles and angle bisector
  • 5. Video lesson on Angles and Angle Bisector.
  • 6. Chapter 2 exercises

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3.

Triangles

The purpose of this chapter is to provide conditions for students to develop the following ability: - Understanding of the definitions and properties related to triangles. In order to do so, we will be exploring activities that allow the improvement of the following skills: • Knowing what a triangle is, by recognizing its different representations and elements (vertices, sides and angles). • Knowing how to classify the triangle in terms of sides: Equilateral, Isosceles, Scalene. • Knowing how to classify the triangle in terms of angles: Acute Angle, Right Angle and Obtuse Angle. • Understanding the Triangle Inequality Theorem.

  • 1. Triangles: Definition and Elements
  • 2. Triangles- Classification
  • 3. Triangular Inequality – building the triangle
  • 4. Video classes about triangles
  • 5. CHAPTER 3 EXERCISES

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4.

Congruence of Triangles

The purpose of this chapter is to provide conditions for students to develop the following skill: - Understanding of the congruence of triangles and the properties related to it. In order to do so, we will be exploring activities that allow the improvement of the following skills: • Understanding of the definition “Congruence of Triangles”. • Knowing and applying the cases of congruence. • Understanding of the proofs of congruence cases. • Understanding and knowing of how to apply the Exterior angle theorem.

  • 1. Congruence of Triangles
  • 2. Triangle Congruence Cases
  • 3. Why does the ASS (or SSA) can't be used to determine triangle congruence?
  • 4. Solved Exercises
  • 5. Video lesson on congruence of triangles
  • 6. Chapter 4 exercises

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5.

Parallelism and Perpendicularity

The purpose of this chapter is provide conditions for students to develop the following skill: • Understanding of the definitions and properties of parallelism and perpendicularity. Certain activities will be explored in order to develop the following skills: • Knowing of what are parallel and transversal lines, and alternate, corresponding and consecutive angles. • Understanding and knowing how to apply the parallelism theorem. • Knowing how to build parallel lines, using a ruler and a compass. • Understanding and knowing how to apply the postulate of “Unicity of the parallel line”. • Understanding and knowing how to apply the Exterior angle theorem. • Understanding and knowing how to apply the theorem "sum of the internal angles of a triangle". • Identifying and knowing how to apply the properties for angles on parallel lines. • Knowing of what are perpendicular and oblique lines. • Knowing how to build perpendicular lines, using a ruler and a compass.

  • 1. Parallelism
  • 2. Perpendicularity
  • 3. Video lessons on parallelism and perpendicularity
  • 4. Chapter 5 exercises

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The Classical Triangle Centers

The purpose of this chapter is to provide students the conditions to develop the following skill: • Understanding of the definitions and properties of the Notable Points of the triangle. In order to do so, we will be exploring activities that allow the improvement of the following skills: • Knowing what is the median of a triangle. • Knowing what is the triangle's barycenter (or centroid) and how to determine it. • Knowing and determining how to apply the property of the medians of the triangle. • Knowing what is the incenter of a triangle and how to determine it. • Knowing what is the triangle's barycenter (or centroid) and how to determine it. • Knowing what is the orthocenter of a triangle and how to determine it. • Knowing the conditions that determine the positions (internal, external, etc.) of the notable points. • Knowing what is the Euler line.

  • 1. Incenter
  • 2. Circumcenter
  • 3. The Orthocenter
  • 4. Barycenter
  • 5. The Euler line and the center of mass of the triangle (centroid)
  • 6. Video lessons on notable points
  • 7. Chapter 6 exercises

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Quadrilaterals - Definitions and Properties

The purpose of this chapter is to provide conditions for students to develop the following skill: • Understanding of the definitions and properties related to the Notable Quadrilaterals (Convex quadrilaterals that have at least two paralel sides). In order to do so, we will be exploring activities that allow the improvement of the following skills: • Knowing of what are concave and convex quadrilaterals. • Understanding and knowing how to apply trapezoid definitions and properties. • Knowing how to classify the different trapezoids (isosceles, scalene and right angle). • Understanding and knowing how to apply the definitions and properties of the parallelogram. • Understanding and knowing how to apply the definitions and properties of the rectangle, rhombus (lozenge) and square. • Understanding and knowing how to apply the Midpoint (Midline or Midsegment) Theorem of the triangle and trapezoid.

  • 1. Quadrilateral-Basic Notions
  • 2. Trapezoid
  • 3. Parallelogram
  • 4. Rectangle
  • 5. Rhombus
  • 6. Square
  • 7. Triangle Midsegment
  • 8. Trapezoid Midsegment
  • 9. Video Lessons about Quadrilaterals (put English subtitles)
  • 10. Chapter 7 - Exercises

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8.

Thales' Theorem and Angle Bisectors' Theorem

The purpose of this chapter is to provide students the conditions to develop the following skill: • Understanding of the definitions and properties related to the Thales Theorem. In order to do so, we will be exploring activities that allow the improvement of the following skills: • Knowing of historical facts related to the Thales Theorem. • Knowing what are sets of parallel lines, transversal lines to sets of parallel ones, and corresponding points and segments. • Understanding and knowing how to prove the Tales theorem. • Understanding and knowing how to prove the interior and exterior bisector theorems.

  • 1. Thales' Theorem
  • 2. Interior Angle Bisector Theorem
  • 3. Exterior angle bisector
  • 4. Dynamic Solved Exercise
  • 5. Video lesson on the Thales theorem
  • 6. Exercises from Chapter 8

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Circumference, Circle and Angles in the Circumference

The purpose of this chapter is to provide students the conditions to develop the following skill: [*]Understanding of the definitions and properties related to circles and circumferences. In order to do so, we will be exploring activities that allow the improvement of the following skills: [*] Knowing the concept of circle and circunference by recognizing their different representations and elements (radius and center). [*] Knowing the concept of the following elements: arc, capable arc, center angle, inscribed angle, segment angle, internal and external eccentric angle. [*] Knowing and applying the properties related to arco capaz, central angle, inscribed angle, segment angle, inscribed quadrilateral, inner and outer eccentric angle.

  • 1. Circumference and Circle
  • 2. Congruent Arcs and Arcs Addition
  • 3. Central Angle, Inscribed Angle, Circular Segment Angle and Arc
  • 4. Inscribed quadrilateral and eccentric angles
  • 5. Video Lessons - Definitions, Relative Positions, etc.
  • 6. Chapter 9 Exercises

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10.

Triangle Similarity, Power of a point, and Metric Relations in the Right Triangle

The purpose of this chapter is to provide conditions for students to develop the following skills: • Understanding of the similarity of triangles and their related properties. • Understanding of the metric relations in the right triangle. In order to do so, we will be exploring activities that allow the improvement of the following skills: • Understanding of the definition of "Triangle Similarity". • Knowing and applying the cases of similarity. • Understanding of the demonstrations of similarity cases. • Understanding and knowing of how to apply point potency properties. • Knowing of how to identify the elements of the right triangle. • Knowing of how to deduce and apply the metric relations of the right triangle. • Understanding of the different demonstrations of the Pythagorean theorem. • Knowing of different applications of Pythagorean theorem: diagonal of a square, height of equilateral triangle, etc.

  • 1. Triangle similarity: Definition, criteria and demonstrations
  • 2. Power of a point
  • 3. Right-Triangle Elements, Metric Relations and Pythagorean Theorem
  • 4. Videoaulas sobre Semelhança de triângulos, Relações métricas no triângulo retângulo e Potência de Ponto
  • 5. Exercícios do capítulo 10

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Area of Rectangle, Square, Parallelogram, Trapezoid, Rhombus, Triangles and Regular Polygons.

The purpose of this chapter is to provide students the conditions to develop the following skill: • Understanding of the measure of the area of plane shapes. In order to do so, we will be exploring activities that allow the improvement of the following skills: • Understanding the concept of equivalence between polygons. • Understanding and knowing how to apply the formula of the rectangle area. • Understanding the reasoning, knowing and applying the formulas for the areas of the square, triangle, parallelogram, trapezoid and rhombus (a lozenge is one type of rhombus).

  • 1. Rectangle Area
  • 2. Square Area
  • 3. Parallelogram Area
  • 4. Triangle Area
  • 5. Trapezoid Area
  • 6. Rhombus Area
  • 7. Regular Polygon Area
  • 8. Exercícios do Capítulo 11

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Circle

The purpose of this chapter is provide conditions for students to develop the following skill: • Understanding of the procedures for calculating the perimeter of the circumference and the area of the circle. In order to do so, we will be exploring activities that allow the improvement of the following skills: • Knowing how to differentiate between circle and circumference. • Understanding of the deduction of the formula for calculating the length of the circumference. • Understanding of the deduction of the formula for calculating the area of the circle. • Knowing of how to calculate the circumference length • Knowing of how to calculate the area of the circle.

  • 1. Length of the Circumference
  • 2. Area of a Circle
  • 3. Video lesson: Circumference and Circle
  • 4. Chapter 12 exercises

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Geometric Drawing

  • 1. Point, Point of Intersection and Midpoint
  • 2. Tracing Angles and Angles in Radians
  • 3. Parallel, Perpendicular, Line Bisector and Angle Bisector
  • 4. Regular Polygons
  • 5. Videoaulas sobre polígonos regulares
  • 6. Exercícios do capítulo 13

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References

  • 1. Bibliographic References and Sites

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Information