Illustrations Without Words

Tim Brzezinski, Allison Spencer, Jun 30, 2016

Contains many dynamic illustrations WITHOUT words, segment lengths, and angle measures. In these applets, the slider governs all applet dynamics. These applets can be used to help a student informally & visually discover a mathematics concept and/or meaningfully recall a certain concept. Enjoy!

Table of Contents

  1. Writing Prompts: Definitions & Theorems
    1. Congruent Angles: Definition
    2. Congruent Segments: Quick Exploration
    3. Midpoint Definition
    4. Perpendicular Bisector Definition
    5. Angle Bisector Definition (I)
    6. Complementary Meaning?
    7. Supplementary Angles (Quick Exploration)
    8. A Special Theorem: Part 2 (V2)
    9. A Special Theorem: Part 2 (V3)
    10. A Special Theorem: Part 2 (V4)
    11. Another Special Theorem: Part 1 (V2)
    12. Another Special Theorem: Part 1 (V3)
  2. Parallel Lines & Related Angles
    1. Vertical Angles Theorem
    2. Exploring Corresponding Angles (V2)
    3. Alternate Interior Angles Theorem (V1)
    4. Exploring Alternate Interior Angles (V2)
    5. Alternate Interior Angles Theorem (V3)
    6. Alternate Exterior Angles Theorem (V1)
    7. Exploring Alternate Exterior Angles (V2)
    8. Alternate Exterior Angles Theorem (V3)
    9. Same Side Interior Angles Theorem
    10. Exploring Same Side Exterior Angles
  3. Polygons & Angles: Interior & Exterior Angles
    1. Triangle Angle Theorems
    2. Triangle Angle Theorems (V2)
    3. Triangle Angle Theorems (V3)
    4. Quadrilateral Angle Theorems
    5. Pentagon Angle Theorems
    6. Hexagon Angle Theorems
    7. Heptagon Angle Theorems
    8. Octagon Angle Theorems
  4. Polygons: Exterior Angles Only
    1. Exterior Angles of a Triangle
    2. Exterior Angles of a Quadrilateral
    3. Exterior Angles of a Pentagon
    4. Exterior Angles of a Hexagon
    5. Exterior Angles of a Heptagon
    6. Exterior Angles of an Octagon
  5. Quadrilaterals: Theorems & Properties
    1. Parallelograms (I)
    2. Parallelogram Diagonals
    3. Quad Midpoints Action!
    4. Rhombus Action!
    5. Rhombus Diagonals
    6. Rectangle Action!
    7. Rectangle Diagonals
    8. Square Action!
    9. Isosceles Trapezoid Action!
    10. Trapezoid Median: 2 Discoveries
    11. Isosceles Trapezoid Diagonals
    12. Isosceles Trapezoid Midpoints
    13. Kite Action!
    14. SSSS: Similar Quads?
    15. AAA: Similar Quads?
    16. Trapezoid Median (Midsegment) Action!
    17. Thebault Action!!!
  6. Triangles: Theorems & Properties
    1. Similar Right Triangles (V1)
    2. AA Similarity Theorem
    3. SAS ~ Theorem
    4. SSS ~ Theorem (V1)
    5. SSS ~ Theorem (V2)
    6. Triangle Midsegment
    7. Parallel Lines Proportionality Theorem
    8. Triangle-Angle Bisector Theorem
    9. A Special Triangle & Its Properties (I)
    10. Isosceles Triangle Medians
    11. Triangle Midsegment Action!
    12. Another Special Triangle and its Properties (II)
    13. Special Right Triangle (I)
    14. Another Special Right Triangle (Guided Discovery)
    15. Vivani Action!
    16. Napoleon Motion!
    17. Incenter & Incircle Action!
    18. Medians and Centroid Dance
    19. Circumcenter & Circumcircle Action!
    20. 3 Special Points!
    21. 9 Point Circle Action
    22. 9-Point Circle Action (Part 2)
    23. 9-Point Circle Action (Part 4)
    24. 9-Point Circle Action (Part 3B)
    25. Converse of IST (V1)
    26. Converse of IST (II)
  7. Circle Theorems
    1. Properties of Tangents Drawn to Circles (A)
    2. Properties of Tangents Drawn to Circles (B)
    3. Diameters & Chords I (VA)
    4. Diameters & Chords I (VB)
    5. Congruent Chords (I)
    6. Inscribed Angle Theorem (V1)
    7. Inscribed Angle Theorem: Corollary 1
    8. Where to Sit? (I)
    9. Thales' Theorem (VB)
    10. Inscribed Angle Intercepts Semicircle
    11. Cyclic Quadrilaterals (IAT: Corollary 3)
    12. Angle Formed by Chord & Tangent (I)
    13. Angle Formed by Chord & Tangent (II)
    14. Angle Formed by 2 Chords (I)
    15. Angle Formed by 2 Chords (II)
    16. Crossing Chords: Proof Hint
    17. Angles from Secants and Tangents (V1)
    18. Angle From 2 Secants (V2)
    19. Secants: Proof Hint
  8. Area, Surface Area, Volume
    1. Parallelogram: Area
    2. Area of a Triangle (Discovery)
    3. Area of a Trapezoid (Discovery)
    4. Right Circular Cylinder!
  9. Conic Sections: Definitions & Properties
    1. Parabola as a Locus
    2. Parabola: Geometric Property (I)
    3. Parabola: Geometric Property (II)
    4. Ellipse: Magic Sum = ?
    5. Ellipse: Magic Sum = ? (V2)
    6. Ellipse: Reflective Property
    7. Hyperbola: Difference = ?
    8. Hyperbola: Reflective Property
  10. Older Versions of Some Proofs in First Few Chapters
    1. Triangle Interior & Exterior Angle Sum Theorems (II)
    2. Quadrilateral Interior & Exterior Angle Sum Theorems (V2)
    3. Quadrilateral Interior & Exterior Angle Sum Theorems (V2)
    4. Pentagon Interior & Exterior Angle Sum Theorems
    5. Hexagon Interior & Exterior Angle Sum Theorems
    6. Heptagon Interior & Exterior Angle Sum Theorems
    7. Octagon Interior & Exterior Angle Sum Theorems
    8. Inscribed Angle Theorem (Intro): What Do You See Here?
    9. Angle formed by a Secant & Tangent: What Do You See? (V1)
    10. Angle Formed by a Secant & Tangent: What Do You See? (V2)
    11. Angle Formed by 2 Tangents (V2): What Do You See?
    12. Cyclic Quadrilaterals
    13. Inscribed Angle Action !!!
    14. Inscribed Angle Action! (2 Theorems)
    15. Angle Formed by 2 Chords (V1): What Do You See?
    16. Angle Formed by 2 Chords (V2): What Do You See?
    17. Similar Right Triangles (II)
    18. Area of a Parallelogram (Discovery)

1.

Writing Prompts: Definitions & Theorems

  • 1. Congruent Angles: Definition
  • 2. Congruent Segments: Quick Exploration
  • 3. Midpoint Definition
  • 4. Perpendicular Bisector Definition
  • 5. Angle Bisector Definition (I)
  • 6. Complementary Meaning?
  • 7. Supplementary Angles (Quick Exploration)
  • 8. A Special Theorem: Part 2 (V2)
  • 9. A Special Theorem: Part 2 (V3)
  • 10. A Special Theorem: Part 2 (V4)
  • 11. Another Special Theorem: Part 1 (V2)
  • 12. Another Special Theorem: Part 1 (V3)

1.1.

Congruent Angles: Definition

[color=#000000]The following applet illustrates what it means for two angles to be classified as congruent angles. [br]Interact with this applet for a few minutes, then answer the questions that follow. [br][i]Be sure to move the [/i][/color][i][b][color=#ff00ff]BIG PINK POINTS[/color][/b][color=#000000] around each time before you re-slide the slider. [/color][/i]
[b][color=#1e84cc]Questions: [/color][/b][color=#000000] [br][br][/color][color=#000000]1) What geometric transformation(s) took place in the applet above? List it/them.[br][br][/color][color=#000000]2) Did any of these transformations change the measure of the [/color][b][color=#6aa84f]original green angle[/color][/b][color=#000000]? [br][br][/color][color=#000000]3) [/color][i][color=#ff0000]Cause:[/color][color=#000000] [br][/color][/i][color=#000000] How would you define what it means for angles to be congruent with respect to the geometric [br] transformations you listed in (1) above? [br][br][/color][color=#000000]4)[/color] [color=#ff0000][i]Effect[/i]:[/color] [br] [color=#000000]If we know that two angles are congruent, what can we consequently conclude about their measures? [/color]
– Tim Brzezinski

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1.7.

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1.8.

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1.9.

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1.10.

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1.11.

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1.12.

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2.

Parallel Lines & Related Angles

  • 1. Vertical Angles Theorem
  • 2. Exploring Corresponding Angles (V2)
  • 3. Alternate Interior Angles Theorem (V1)
  • 4. Exploring Alternate Interior Angles (V2)
  • 5. Alternate Interior Angles Theorem (V3)
  • 6. Alternate Exterior Angles Theorem (V1)
  • 7. Exploring Alternate Exterior Angles (V2)
  • 8. Alternate Exterior Angles Theorem (V3)
  • 9. Same Side Interior Angles Theorem
  • 10. Exploring Same Side Exterior Angles

2.1.

Vertical Angles Theorem

[b]Definition:[/b] [b][color=#b20ea8]Vertical Angles[/color][/b] are angles whose sides form 2 pairs of opposite rays. [br][br]When 2 lines intersect, 2 pairs of vertical angles are formed. [color=#b20ea8]One pair of vertical angles is shown below. [/color] [br][color=#888](Click the other checkbox on the right to display the other pair of vertical angles.) [/color][br][br]Interact with the following applet for a few minutes, then answer the questions that follow.
Directions & Questions: [br][br]1) Complete the following statement (based upon your observations). [br] [br] [color=#b20ea8][b]Vertical angles are always __________________________.[/b] [/color] [br][br]2) Suppose the pink angle above measures 140 degrees. What would the measure of its vertical angle? What would be the measure of the other 2 (gray) angles?
– Tim Brzezinski

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2.7.

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2.8.

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2.9.

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2.10.

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3.

Polygons & Angles: Interior & Exterior Angles

  • 1. Triangle Angle Theorems
  • 2. Triangle Angle Theorems (V2)
  • 3. Triangle Angle Theorems (V3)
  • 4. Quadrilateral Angle Theorems
  • 5. Pentagon Angle Theorems
  • 6. Hexagon Angle Theorems
  • 7. Heptagon Angle Theorems
  • 8. Octagon Angle Theorems

3.1.

Triangle Angle Theorems

Interact with the app below for a few minutes.  [br]Then, answer the questions that follow.  [br][br]Be sure to change the locations of this triangle's vertices each time [i]before[/i] you drag the slider!
What is the [b]sum of the measures of the interior angles of this triangle? [/b]
What is the [b]sum of the measures of the exterior angles [/b]of this triangle?
– Tim Brzezinski

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4.

Polygons: Exterior Angles Only

  • 1. Exterior Angles of a Triangle
  • 2. Exterior Angles of a Quadrilateral
  • 3. Exterior Angles of a Pentagon
  • 4. Exterior Angles of a Hexagon
  • 5. Exterior Angles of a Heptagon
  • 6. Exterior Angles of an Octagon

4.1.

Exterior Angles of a Triangle

– Tim Brzezinski

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5.

Quadrilaterals: Theorems & Properties

  • 1. Parallelograms (I)
  • 2. Parallelogram Diagonals
  • 3. Quad Midpoints Action!
  • 4. Rhombus Action!
  • 5. Rhombus Diagonals
  • 6. Rectangle Action!
  • 7. Rectangle Diagonals
  • 8. Square Action!
  • 9. Isosceles Trapezoid Action!
  • 10. Trapezoid Median: 2 Discoveries
  • 11. Isosceles Trapezoid Diagonals
  • 12. Isosceles Trapezoid Midpoints
  • 13. Kite Action!
  • 14. SSSS: Similar Quads?
  • 15. AAA: Similar Quads?
  • 16. Trapezoid Median (Midsegment) Action!
  • 17. Thebault Action!!!

5.1.

Parallelograms (I)

[color=#000000]Please use these applets to help you complete the [/color][i][color=#0000ff]Parallelogram Investigation[/color][/i][color=#000000] questions given to you at the beginning of class. [/color]
Sides of a Parallelogram
Interior Angles of a Parallelogram
Diagonals of a Parallelogram
– Tim Brzezinski

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5.7.

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5.10.

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5.11.

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5.12.

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5.13.

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5.14.

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5.15.

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5.16.

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5.17.

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6.

Triangles: Theorems & Properties

  • 1. Similar Right Triangles (V1)
  • 2. AA Similarity Theorem
  • 3. SAS ~ Theorem
  • 4. SSS ~ Theorem (V1)
  • 5. SSS ~ Theorem (V2)
  • 6. Triangle Midsegment
  • 7. Parallel Lines Proportionality Theorem
  • 8. Triangle-Angle Bisector Theorem
  • 9. A Special Triangle & Its Properties (I)
  • 10. Isosceles Triangle Medians
  • 11. Triangle Midsegment Action!
  • 12. Another Special Triangle and its Properties (II)
  • 13. Special Right Triangle (I)
  • 14. Another Special Right Triangle (Guided Discovery)
  • 15. Vivani Action!
  • 16. Napoleon Motion!
  • 17. Incenter & Incircle Action!
  • 18. Medians and Centroid Dance
  • 19. Circumcenter & Circumcircle Action!
  • 20. 3 Special Points!
  • 21. 9 Point Circle Action
  • 22. 9-Point Circle Action (Part 2)
  • 23. 9-Point Circle Action (Part 4)
  • 24. 9-Point Circle Action (Part 3B)
  • 25. Converse of IST (V1)
  • 26. Converse of IST (II)

6.1.

Similar Right Triangles (V1)

[color=#000000]Interact with the applet below for a few minutes, then answer the questions that follow.[/color]
1.
[color=#000000]What is the sum of the measures of the [/color][color=#ff0000]red[/color] [color=#000000]and[/color] [color=#0000ff]blue[/color] [color=#000000]angles?  [br]How do you know this to be true?  [/color]
2.
The segment that was drawn as you dragged the slider is called an [b]altitude.[br][/b]This [b]altitude [/b]was [b]drawn to the hypotenuse[/b].  [b]  [br][/b]How many right triangles did this [b]altitude[/b] split the original right triangle into?
3.
[color=#000000]What does the the special movement of the [/color][color=#ff0000]red[/color][color=#000000] and [/color][color=#0000ff]blue[/color][color=#000000] angles imply about[br]these 2 smaller right triangles?  [br][br]What previously learned theorem justifies your answer?  [/color]
4.
Does your response for (3) also hold true for the relationship between the ORIGINAL[br]BIG RIGHT TRIANGLE and either one of the smaller right triangles?[br][br]If so, how/why do you know this?  
Quick (Silent) Demo
– Tim Brzezinski

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6.11.

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6.12.

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6.14.

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6.15.

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6.17.

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6.18.

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6.19.

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6.23.

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7.

Circle Theorems

  • 1. Properties of Tangents Drawn to Circles (A)
  • 2. Properties of Tangents Drawn to Circles (B)
  • 3. Diameters & Chords I (VA)
  • 4. Diameters & Chords I (VB)
  • 5. Congruent Chords (I)
  • 6. Inscribed Angle Theorem (V1)
  • 7. Inscribed Angle Theorem: Corollary 1
  • 8. Where to Sit? (I)
  • 9. Thales' Theorem (VB)
  • 10. Inscribed Angle Intercepts Semicircle
  • 11. Cyclic Quadrilaterals (IAT: Corollary 3)
  • 12. Angle Formed by Chord & Tangent (I)
  • 13. Angle Formed by Chord & Tangent (II)
  • 14. Angle Formed by 2 Chords (I)
  • 15. Angle Formed by 2 Chords (II)
  • 16. Crossing Chords: Proof Hint
  • 17. Angles from Secants and Tangents (V1)
  • 18. Angle From 2 Secants (V2)
  • 19. Secants: Proof Hint

7.1.

Properties of Tangents Drawn to Circles (A)

In the applet below, 2 tangent rays are drawn to a circle from a point outside that circle. [br][br]Interact with the applet below for a few minutes. As you do, be sure to change the locations of the [b][color=#3c78d8]blue points[/color][/b] (i.e. alter the size of the circle) and [b][color=#bf9000]yellow point[/color][/b]. [br] [br]After doing so, please answer the questions that follow.
1.
What is the measure of each [b][color=#ff00ff]pink angle[/color][/b] displayed? Explain how you know this.
2.
What can you conclude about the [b]distances[/b] from the [b][color=#bf9000]yellow point (outside the circle)[/color][/b] to each of the smaller white points on the circle?
Quick (Silent) Demo
3.
Given your response for (1), how could we prove that your response to (2) is true? (Assume we didn't see the rotation at the very end of the animation).
– Tim Brzezinski

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7.10.

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7.11.

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7.12.

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7.13.

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7.14.

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7.16.

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7.17.

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7.18.

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7.19.

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8.

Area, Surface Area, Volume

  • 1. Parallelogram: Area
  • 2. Area of a Triangle (Discovery)
  • 3. Area of a Trapezoid (Discovery)
  • 4. Right Circular Cylinder!

8.1.

Parallelogram: Area

Directions:
In the app below, use the [b]filling[/b] slider to make the parallelogram light enough so that you can see the white gridlines through it. [b]But don't touch anything else yet! [/b][br][br]After you set the [b]filling[/b] slider, try to count the number of squares inside this parallelogram. [br]Be sure to include partial squares! Provide a good estimate in the question box below.
Try to count the number of squares inside this parallelogram. How many squares (i.e. square units) do you estimate to be inside this parallelogram?
Slide the "[b]Slide Me" [/b]slider now. Carefully observe what happens. What shape do you see now?
How does the area of this new shape compare with the area of the original parallelogram? How do you know this?
How many squares do you now count in the newer shape that was formed? How many squares were in the original parallelogram?
Without looking it up on another tab in your browser, describe how we can find the area of ANY PARALLELOGRAM.
– Tim Brzezinski

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9.

Conic Sections: Definitions & Properties

  • 1. Parabola as a Locus
  • 2. Parabola: Geometric Property (I)
  • 3. Parabola: Geometric Property (II)
  • 4. Ellipse: Magic Sum = ?
  • 5. Ellipse: Magic Sum = ? (V2)
  • 6. Ellipse: Reflective Property
  • 7. Hyperbola: Difference = ?
  • 8. Hyperbola: Reflective Property

9.1.

Parabola as a Locus

Here, we have a parabola with its focus and directrix. Point P lies on the parabola itself. Mess around with this app by moving any of the BIG POINTS anywhere you'd like.
Other than mentioning the fact that [color=#bf9000][b]P[/b] [/color]always lies on a parabola, how would you describe all possible locations for [b][color=#bf9000]P[/color][/b] above? Be specific.
– Tim Brzezinski

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10.

Older Versions of Some Proofs in First Few Chapters

  • 1. Triangle Interior & Exterior Angle Sum Theorems (II)
  • 2. Quadrilateral Interior & Exterior Angle Sum Theorems (V2)
  • 3. Quadrilateral Interior & Exterior Angle Sum Theorems (V2)
  • 4. Pentagon Interior & Exterior Angle Sum Theorems
  • 5. Hexagon Interior & Exterior Angle Sum Theorems
  • 6. Heptagon Interior & Exterior Angle Sum Theorems
  • 7. Octagon Interior & Exterior Angle Sum Theorems
  • 8. Inscribed Angle Theorem (Intro): What Do You See Here?
  • 9. Angle formed by a Secant & Tangent: What Do You See? (V1)
  • 10. Angle Formed by a Secant & Tangent: What Do You See? (V2)
  • 11. Angle Formed by 2 Tangents (V2): What Do You See?
  • 12. Cyclic Quadrilaterals
  • 13. Inscribed Angle Action !!!
  • 14. Inscribed Angle Action! (2 Theorems)
  • 15. Angle Formed by 2 Chords (V1): What Do You See?
  • 16. Angle Formed by 2 Chords (V2): What Do You See?
  • 17. Similar Right Triangles (II)
  • 18. Area of a Parallelogram (Discovery)

10.1.

Triangle Interior & Exterior Angle Sum Theorems (II)

[b][color=#c51414]Interact with the applet below for a few minutes.[br]Answer the questions that follow.[/color][br][/b]
[b][color=#c51414]Questions:[/color][/b][br][br]1) What can you conclude about a triangle's interior angle measures? [i]Be specific.[/i][br]2) What can you conclude about a triangle's exterior angle measures? [i]Be specific.[/i][br][br]3) [b]Use [/b][b][color=#c51414]transformational geometry[/color][/b] t[b]o prove your conclusions for both (1) and (2) are correct. [/b]
– Tim Brzezinski

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10.17.

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10.18.

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Information