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Geometry Collected Resources

Dr. Jack L. Jackson II, Jan 31, 2022

This GeoGebra book collects activities for use in the study of Geometry. It includes all of the public activities written by Dr. Jackson along with some of the best activities written and made publicly available by other mathematics educators. This collection includes activities used with Dr. Jackson's College Geometry course. In that course, he works simultaneously in Taxicab, Hyperbolic, and Spherical Geometry along with the traditional Euclidean Geometry. In particular, Dr. Jackson has created two activities: Spherical Geometry Environment and Unified Geometry Environment. The Spherical Geometry Environment has custom tools for working in Spherical Geometry. The Unified Geometry Environment has tools for working in Euclidean, Taxicab, and Hyperbolic Geometries. The model used for Hyperbolic Geometry is the Beltrami-Poincare Half-plane model. These environments have non-Euclidean versions of all of the built-in Euclidean Geometry tools. Be sure to open them in the app to have access to the tools. Because of the number of custom tools, these files may open slowly. While this collection includes some activities for a more advanced course, most of these activities are also appropriate for a high school course in Geometry and some are appropriate for elementary and middle level students..

1.

1. Some GeoGebra Basics

These are some introductory resources for getting started with GeoGebra and learning how to use using basic tools.

1. Drawing vs. Constructing, The Drag Test

1.1.

2.

2. Basic Unified Geometry Environments

These files provide environments for Spherical, Hyperbolic, Taxicab, and Euclidean Geometry. They include custom tools for working with the non-Euclidean geometries in the same way that the built-in tools work with Euclidean Geometry. Because of the number of custom tools, these files may open slowly. Be sure to open them in the app so that the tools become available.

1. Unified Geometry Environment
2. Spherical Geometry Environment
3. Unified Geometry Environment B

2.1.

2.2.

2.3.

3.

3. Introducing Multiple Geometries

We introduce models for several geometries including several finite geometries and the four main infinite geometries that we study as part of Unified Geometry: Euclidean, Hyperbolic (half-plane model), Spherical, and Taxicab Geometries. We see how to model basic objects such as points, lines, line segments, rays, angles, and planes. We use custom tools for the non-Euclidean geometries. We also examine the most basic postulates and properties of incidence.

1. Spherical Geometry Environment
2. Unified Geometry Environment
3. Hyperbolic Geometry: The Poincare Disk Model

3.1.

3.2.

3.3.

4.

4. Distance and Order

We focus on properties of distance and order in multiple geometries. We see the distance metric in Euclidean, Hyperbolic, Spherical, and Taxicab Geometries.

1. Basic Objects
2. Naming Points, Segments, Rays, and Lines
3. Naming Geometric Objects
4. Centimeter Ruler: Custom Templates
5. Quiz: Distance in the Coordinate Plane (V1)
6. Euclidean and Taxicab Distance
7. AQR Section 37: Taxicab Distance v. Euclidean Distance
8. Hyperbolic Distance Metric

4.1.

4.2.

4.3.

4.4.

4.5.

4.6.

4.7.

4.8.

5.

5. Angles

We introduce angles and angle measure in multiple geometries.

1. Discovering Ways to Name Angles
2. Naming Angles
3. Naming Angles According to Their Size
4. Measuring Angles
5. What is a Radian?
6. Identifying Angles Around Us
7. Guess the Radian Measure
8. Drawing Angles Game Using Radians
9. Determining Angle Measures Without Tools
10. Drawing Angles: Hot and Cold Activity
11. Measuring Angles Using a Protractor
12. Drawing Angles Using a Protractor
13. Converting Degrees to Radians
14. Drawing Angles Game
15. Drawing Angles: Hot and Cold Activity
16. Creating Angles on a Grid
17. Degrees to Radians and Back
18. Congruent Angles: Definition
19. Vertical Angles Exploration (2)
20. Vertical Angles Theorem
21. Vertical Angles
22. Adjacent Angles
23. Finding the Missing Angle
24. Estimating Angle Measures Using Radians
25. Determining Angle Measures Without Tools
26. Complementary and Supplementary Angles
27. Unknown Complementary Angles
28. Supplementary Angles (Quick Exploration)
29. Drawing Parallel, Perpendicular, and Intersecting Lines
30. Adding Adjacent Angles

5.1.

5.2.

5.3.

5.4.

5.5.

5.6.

5.7.

5.8.

5.9.

5.10.

5.11.

5.12.

5.13.

5.14.

5.15.

5.16.

5.17.

5.18.

5.19.

5.20.

5.21.

5.22.

5.23.

5.24.

5.25.

5.26.

5.27.

5.28.

5.29.

5.30.

6.

6.A. Triangles in Euclidean Geometry

We explore properties of Triangles in Euclidean Geometry, including classification of triangles, the sum of the measures of the interior angles, and congruence criteria.

1. Euclidean General Triangle measured
2. Classifying Triangles in Euclidean Geometry
3. Exploring Types of Triangles
4. Relational Parts of a Triangle
5. Drawing Triangles
6. Isosceles Triangles in Euclidean,. Hyperbolic, and Taxicab
7. Definition of Congruent Triangles
8. Triangle Congruence
9. SAS Triangle Congruence Exploration Euclidean Geometry
10. SAS: Dynamic Proof!
11. ASA Triangle Congruence Exploration Euclidean Geometry
12. ASA Triangle Congruence Theorem Proof
13. ASA Theorem?
14. SSS Triangle Congruence Exploration Euclidean Geometry
15. SSS: Dynamic Proof!
16. Drawing Triangles
17. Visualizing Triangles
18. Compass and Straightedge Constructions
19. Exterior Angle Inequality Proof with Euclidean Illustration
20. Triangle Exterior Angle
21. AAS Triangle Congruence Exploration Euclidean Geometry
22. Angle-Angle-Side (AAS): Quick Exploration
23. SSA Triangle Congruence Exploration Euclidean Geometry
24. SSA Theorem?
25. AAA Triangle Exploration Euclidean Geometry
26. Sum of Interior Angles of a Triangle
27. Triangle Angle Theorems
28. Triangle Solver
29. Open Middle: Triangle Angles
30. Shortest Distance Theorem
31. Triangle Centers (Euclidean and Hyperbolic Geometry)
32. Triangle Fact (I)
33. 9 Point Circle Action
34. Pythagorean Animation
35. Pythagorean Theorem Animation
36. Finding the Distance Between Two Points Using the Pythagorean Theorem
37. Identifying Right Triangles Using the Pythagorean Theorem
38. Sum of Interior Angles of a Triangle

6.1.

6.2.

6.3.

6.4.

6.5.

6.6.

6.7.

6.8.

6.9.

6.10.

6.11.

6.12.

6.13.

6.14.

6.15.

6.16.

6.17.

6.18.

6.19.

6.20.

6.21.

6.22.

6.23.

6.24.

6.25.

6.26.

6.27.

6.28.

6.29.

6.30.

6.31.

6.32.

6.33.

6.34.

6.35.

6.36.

6.37.

6.38.

7.

6b. Triangles in Hyperbolic Geometry

These activities are exploring properties of triangles in the Half-plane model of Hyperbolic Geometry. Among other things, they explore the sum of the interior angles of triangles, and congruence criteria.

1. Hyperbolic General Tiangle Measured
2. SAS Exploration Hyperbolic Geometry
3. ASA Exploration Hyperbolic Geometry
4. Exterior Angle Inequality Proof Hyperbolic Geometry
5. AAS Exploration Hyperbolic Geometry
6. SSS Investigation Hyperbolic Geometry
7. SSA Exploration Hyperbolic Geometry
8. AAA Exploration Hyperbolic Geometry
9. Triangle Centers (Euclidean and Hyperbolic Geometry)

7.1.

7.2.

7.3.

7.4.

7.5.

7.6.

7.7.

7.8.

7.9.

8.

6.C. Triangles in Spherical Geometry

We explore triangles in Spherical Geometry, including the sum of the measures of the interior angles, and congruence criteria.

1. General Triangle in Spherical Geometry
2. Isosceles Triangle in Spherical Geometry
3. SAS Exploration Spherical Geometry
4. ASA Exploration Spherical Geometry
5. Exterior Angle Inequality Spherical Geometry
6. AAS Exploration Spherical Geometry
7. AAS, AASS Counterexample in Spherical Geometry
8. SSS Exploration Spherical Geometry
9. SSA Exploration Spherical Geometry
10. AAA Exploration Spherical Geometry

8.1.

8.2.

8.3.

8.4.

8.5.

8.6.

8.7.

8.8.

8.9.

8.10.

9.

6.D. Triangles in Taxicab Geometry

We explore triangles in Taxicab Geometry, including the sum of the measures of the interior angles, and congruence criteria.

1. Taxicab General Triangle
2. Euclidean Rotation in Taxicab Geometry
3. SAS Exploration Taxicab Geometry
4. ASA and AAS Exploration Taxicab Geometry
5. SSS Exploration Taxicab Geometry
6. SSA Exploration Taxicab Geometry
7. AA(A) Exploration Taxicab Geometry
8. Taxicab Congruent Triangle Counterexamples

9.1.

9.2.

9.3.

9.4.

9.5.

9.6.

9.7.

9.8.

10.

7. Parallel Lines

We introduce properties of parallel lines in multiple geometries. In particular, the Transversal to Parallel Lines Theorem is examined.

1. Transversal to Parallel Lines
2. Transversal Intersects Parallel Lines
3. Alternate Interior Angles Theorem (V1)
4. Same Side Interior Angles Theorem
5. Exploring Alternate Exterior Angles (V2)
6. Exploring Same Side Exterior Angles
7. Vertical Angles Formed by Parallel Lines and a Transversal
8. Alternate Angles Formed by Parallel Lines and a Transversal
9. Corresponding Angles Formed By Parallel Lines and a Transversal
10. Supplementary Angles Formed by Parallel Lines and a Transversal
11. Parallel Lines & Related Angles
12. Identifying Parallel Lines Cut by a Transversal

10.1.

10.2.

10.3.

10.4.

10.5.

10.6.

10.7.

10.8.

10.9.

10.10.

10.11.

10.12.

11.

8. Quadrilaterals

We examine quadrilaterals in multiple geometries. We explore a robust hierarchical classification scheme for quadrilateral. We then thoroughly explore the properties of each class of quadrilaterals. We also examine congruence criteria for quadrilaterals.

1. Classification of Quadrilaterals in Euclidean Geometry
2. Quadrilateral Angle Theorems
3. Convex quadrilaterals
4. Classifying Quadrilaterals
5. Drawing Quadrilaterals Using Properties
6. Quadrilaterals
7. Mystery Quadrilaterals
8. Quadrilateral Angle Theorems
9. Convex vs. Concave
10. Quadrilateral Creation Templates
11. Kite Action!
12. Kite Template with Investigation Questions
13. Kite in Hyperbolic Geometry
14. Kites in Spherical Geometry
15. Kites in Taxicab Geometry
16. Parallelogram: Theorem 2
17. Parallelogram: Theorem (3)
18. Parallelograms (I)
19. Trapezoid Median (Midsegment) Action!
20. Trapezoid Median: 2 Discoveries
21. Isosceles Trapezoid Template (Scaffolded Discovery)
22. Isosceles Trapezoid Action!
23. Concave Quadrilateral Craziness! (GoGeometry Action 80)
24. Concave Quadrilateral Craziness! (GoGeometry Action 80)
25. Rhombus Action!
26. Quadrilateral Surprise!
27. Quad Midpoints Action!
28. Isosceles Trapezoid Midpoints
29. Compass and Straightedge Constructions
30. Cyclic Quadrilateral: Proof Hint
31. Cyclic Quadrilaterals (IAT: Corollary 3)
32. Quadrilateral "Guess Who?"
33. Square + Rhombus = Constant Surprise! (GoGeometry Action 71)
34. Rotational Symmetry of Quadrilaterals
35. Japanese Theorem for Cyclic Quadrilaterals
36. Quadrilateral Creation Templates
37. Quadrilateral: Exterior Angles
38. Quadrilateral Surprise!
39. Cyclic Quadrilaterals (IAT: Corollary 3)
40. MIdline Quadrilateral 1

11.1.

11.2.

11.3.

11.4.

11.5.

11.6.

11.7.

11.8.

11.9.

11.10.

11.11.

11.12.

11.13.

11.14.

11.15.

11.16.

11.17.

11.18.

11.19.

11.20.

11.21.

11.22.

11.23.

11.24.

11.25.

11.26.

11.27.

11.28.

11.29.

11.30.

11.31.

11.32.

11.33.

11.34.

11.35.

11.36.

11.37.

11.38.

11.39.

11.40.

12.

9. Polygons

We examine properties of polygons, particularly those with more than 4 sides.

1. Naming Polygons
2. Parts of a Polygon
3. Sum of Interior Angles of a Polygon
4. Finding the Missing Angle of a Polygon
5. Polygons: Exterior Angles - REVAMPED
6. Property of cyclic polygon
7. Pentagon Angle Theorems

12.1.

12.2.

12.3.

12.4.

12.5.

12.6.

12.7.

13.

10. Similarity

The concept of similarity in Euclidean Geometry is examined. Theorems related to similarity are explored.

1. AAA Triangle Exploration Euclidean Geometry
2. SAS Triangle Similarity Theorem
3. SSS Triangle Similarity Theorem
4. Similar Triangles or Not
5. Parallel Lines Proportionality Theorem
6. Similar Right Triangles (V2)
7. Triangle Midsegment Action!
8. Making Similar Triangles
9. Harmonic Mean In an Isosceles Trapezoid
10. Identifying and Measuring Corresponding Sides of Similar Figures
11. Changing Scale Factor
12. Calculating Scale Factors of Similar Figures
13. Corresponding Parts of Similar Figures
14. Similar triangles with missing lengths
15. Building Similar Triangles
16. Building Similar Triangles V2
17. Similar Triangles Investigations
18. Similar Triangles
19. Two Transversals to Multiple Parallel Lines
20. Pythagorean and similar triangles.
21. Triangle Midlines, Medians, and Centroid
22. MIdline Quadrilateral 1
23. 2 Noncongruent Triangles with 5 Congruent Parts

13.1.

13.2.

13.3.

13.4.

13.5.

13.6.

13.7.

13.8.

13.9.

13.10.

13.11.

13.12.

13.13.

13.14.

13.15.

13.16.

13.17.

13.18.

13.19.

13.20.

13.21.

13.22.

13.23.

14.

11. Circle Theorems

We examine several theorems related to circles in Euclidean Geometry.

1. Circle with Related Figures
2. Circumference = ? (Animation)
3. Equidistant Chord Action
4. Equidistant Chords, Tangent Lines
5. Where to Sit? (I)
6. Inscribed Angle
7. Inscribed Angles Investigation (Revamped)
8. Inscribed Angle Theorem (V1)
9. Inscribed Angle Theorem: Corollary 1
10. Theorem of Thales
11. Intersecting Secant Lines to a Circle
12. Angle Formed by 2 Chords (I)
13. Angle From 2 Secants (V2)
14. Crossing Chords: Proof Hint

14.1.

14.2.

14.3.

14.4.

14.5.

14.6.

14.7.

14.8.

14.9.

14.10.

14.11.

14.12.

14.13.

14.14.

15.

12.A. Perimeter, Area

We examine measurement concepts in Euclidean Geometry including length, perimeter, and area. We examine area formulas for several types of geometric figures.