Geometry (Similarity, Right T, Trig.)

Tim Brzezinski, Deirdre Longo, Jun 6, 2018

[b]This GeoGebra Book contains lots of discovery-based learning activities, investigations, and meaningful remediation worksheets that were designed to help enhance students' learning of geometry concepts both inside and outside of the mathematics classroom. [color=#1551b5]This GeoGebra Book also contains contributions from the following individuals: Jennifer Silverman ([url]https://www.geogebra.org/jennifer+silverman[/url]) Steve Phelps ([url]https://www.geogebra.org/stevephelps[/url]) Dr. Ted Coe ([url]https://www.geogebra.org/tedcoe[/url]) Without their amazing talent and creativity, this resource would not have been possible. [/b][/color]

Table of Contents

  1. HSG.SRT.A.1, A.1.A, A.1.B
    1. Dilation Introduction: Transformers!
    2. Exploring Dilations: Ex. 19
    3. Dilating a Point (Intro)
    4. Dilating a Line: HSG.SRT.A.1.A
    5. Dilating a Segment: HSG.SRT.A.1.B
  2. HSG.SRT.A.2
    1. Similar Figures: Dynamic Illustration
    2. Animation 243
    3. Properties of Dilations
    4. Properties of Dilations (II)
    5. Proving Triangles Similar (1)
    6. Proving Triangles Similar (2)
    7. Proving Triangles Similar (3)
    8. Proving Triangles Similar (4)
  3. HSG.SRT.A.3
    1. Angle -Angle: Triangle Investigation
    2. AA Similarity Theorem
    3. Animation 97
    4. AAA: Similar Quads?
  4. HSG.SRT.B.4 & B.5
    1. SAS ~ Theorem
    2. SSS ~ Theorem (V1)
    3. Animation 63
    4. SSS ~ Theorem (V2)
    5. SSSS: Similar Quads?
    6. Similar Right Triangles (V1)
    7. Similar Right Triangles (V2)
    8. Animation 13
    9. Pythagorean and similar triangles.
    10. Effortless Trisections
    11. Animation 134
    12. Parallel Lines Congruent Segments Theorem
    13. Parallel Lines Proportionality Theorem
    14. Triangle-Angle Bisector Theorem
    15. Animation 81
    16. Scaling Figures: Effects on Areas
    17. Proof Without Words
    18. Wordless Pythagorean Theorem based on Translations
    19. Proof Without Words
    20. Proof Without Words
    21. Proof Without Words
    22. Proof Without Words
    23. Proof Without Words
    24. Proof Without Words
    25. Proof Without Words
    26. Proof Without Words
    27. Proof Without Words
  5. HSG.SRT.C.6 & C.7 & C.8
    1. Practice: Pythagorean Theorem (1)
    2. Quiz: Pythagorean Theorem (2B)
    3. Angles in Standard Position
    4. Animation 176
    5. Does R Matter?
    6. Quiz: Evaluating Trigonometric Functions of Angles Given a Point on its Terminal Ray
    7. Evaluating Trig Functions Given a Point on the Terminal Ray
    8. Trig Ratio Possibilities (Hints)
    9. Right Triangle Trig: Solving for Sides (2)
    10. Quiz: Special Right Triangles
    11. Special Angles in Standard Position: Intro (V2)
    12. Special Angles in Standard Position: Intro (V3)
    13. Special Angles in Standard Position: Intro (V1)
    14. Animation 177
    15. Animation 178
    16. Right Triangle Trigonometry: Intro
    17. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
    18. Trig Ratio Estimations (Right Triangle Context)
    19. Trig Ratio Estimations (Quiz: V1)
    20. Trig Ratio Estimations (Quiz: V2)
    21. Trigonometric Ratios (Right Triangle Context)
    22. Discovering Sine, Cosine, and Tangent in Right Triangles
    23. Special Trigonometric Identities: Quick Exploration
    24. What "CO" in COsine Means
    25. Quiz: Expressing Angles Using Inverse Trigonometric Function
    26. Animation 98
    27. Animation 99
    28. Animation 100
    29. YOUR Linear Speed?
    30. How Fast are You Spinning?
  6. HSG.SRT.D.9 & D.10 & D.11
    1. Triangle Investigation
    2. Law of Sines (& Area)
    3. Law of Sines (Illustrated)
    4. Law of Cosines: Discovery
    5. Bearing Illustrator (1)
    6. Direction Angle Illustrator
    7. Bearing Illustrator (2)
    8. Bearings Summary
    9. Bearing: Dynamic Illustration
    10. Bearings: Dynamic Illustrator
  7. Links to Other CCSS High School: Geometry Standards
    1. Geometry Resources
  8. Additional - Links to CCSS High School: Functions Resources
    1. Functions Resources

1.

HSG.SRT.A.1, A.1.A, A.1.B

  • 1. Dilation Introduction: Transformers!
  • 2. Exploring Dilations: Ex. 19
  • 3. Dilating a Point (Intro)
  • 4. Dilating a Line: HSG.SRT.A.1.A
  • 5. Dilating a Segment: HSG.SRT.A.1.B

1.1.

Dilation Introduction: Transformers!

Explore this applet to see how the center of dilation and the scale factor affect the image.
Turn off the images and explore the way the lines change. What do you notice?
– Jennifer Silverman

1.2.

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1.3.

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1.4.

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1.5.

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2.

HSG.SRT.A.2

  • 1. Similar Figures: Dynamic Illustration
  • 2. Animation 243
  • 3. Properties of Dilations
  • 4. Properties of Dilations (II)
  • 5. Proving Triangles Similar (1)
  • 6. Proving Triangles Similar (2)
  • 7. Proving Triangles Similar (3)
  • 8. Proving Triangles Similar (4)

2.1.

Similar Figures: Dynamic Illustration

SIMILAR FIGURES
[b]DEFINITION:[br][br]ANY 2 figures are said to be SIMILAR FIGURES if and only if one can be mapped perfectly onto the other under a single transformation OR a composition of 2 or more transformations. [br][br][/b]The applet below dynamically illustrates what it means, by definition, for any 2 triangles to be similar. [br]Feel free to move any of the white vertices anywhere you'd like. [color=#38761d][b]You can also change the size of the green triangle by moving the green slider. [/b][/color]
Quick (Silent) Demo
– Tim Brzezinski

2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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2.7.

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2.8.

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3.

HSG.SRT.A.3

  • 1. Angle -Angle: Triangle Investigation
  • 2. AA Similarity Theorem
  • 3. Animation 97
  • 4. AAA: Similar Quads?

3.1.

Angle -Angle: Triangle Investigation

[b]Students:[br][/b][br]Please use the GeoGebra task applet below to complete the [b]Angle-Angle (Investigation) [/b]given to you at the beginning of class. [br][br][b]LINK:[/b] [b][color=#0000ff][url=https://docs.google.com/document/d/10gizSV-6UM_E6amlcEYLmVmnWgqpW_jWti6MYAyoSwk/edit?usp=sharing]Angle-Angle (Investigation)[/url][/color] [/b]
– Tim Brzezinski

3.2.

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3.3.

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3.4.

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4.

HSG.SRT.B.4 & B.5

  • 1. SAS ~ Theorem
  • 2. SSS ~ Theorem (V1)
  • 3. Animation 63
  • 4. SSS ~ Theorem (V2)
  • 5. SSSS: Similar Quads?
  • 6. Similar Right Triangles (V1)
  • 7. Similar Right Triangles (V2)
  • 8. Animation 13
  • 9. Pythagorean and similar triangles.
  • 10. Effortless Trisections
  • 11. Animation 134
  • 12. Parallel Lines Congruent Segments Theorem
  • 13. Parallel Lines Proportionality Theorem
  • 14. Triangle-Angle Bisector Theorem
  • 15. Animation 81
  • 16. Scaling Figures: Effects on Areas
  • 17. Proof Without Words
  • 18. Wordless Pythagorean Theorem based on Translations
  • 19. Proof Without Words
  • 20. Proof Without Words
  • 21. Proof Without Words
  • 22. Proof Without Words
  • 23. Proof Without Words
  • 24. Proof Without Words
  • 25. Proof Without Words
  • 26. Proof Without Words
  • 27. Proof Without Words

4.1.

SAS ~ Theorem

[color=#000000]In the applet below, you'll find two triangles.  [br][br]The [b]black angle[/b] in the [/color][color=#38761d][b]green triangle[/b][/color] [b][color=#000000]is congruent to[/color][/b][color=#000000] the [/color][b][color=#000000]black angle[/color][/b][color=#000000] in the [/color][b][color=#ff00ff]pink triangle[/color][/b][color=#000000]. [/color][br][br][color=#000000]In the [/color][color=#38761d][b]green triangle[/b][/color][color=#000000], the [b]black angle is the included angle between sides [/b][/color][b][i][color=#000000]a[/color][/i][color=#000000] and [/color][i][color=#000000]b[/color][/i][/b][color=#000000].  [/color][br][color=#000000]In the [/color][b][color=#ff00ff]pink triangle[/color][/b][color=#000000], the [b]black angle is the included angle between sides [i]ka[/i] and [i]kb[/i][/b].  [/color][br][br][color=#000000]Interact with the applet below for a few minutes. [/color][color=#000000]As you do, be sure to move the locations of the [/color][color=#38761d][b]green triangle's[/b][/color][color=#000000] [b]BIG BLACK VERTICES[/b] and the location of the [b]BIG X[/b].[br][/color][color=#000000]You can also adjust the value of [/color][i][color=#000000]k[/color][/i][color=#000000] by using the slider or by entering a value between 0 & 1. [/color][color=#000000] [br][/color][color=#000000] [/color][br] 
[color=#000000]Notice how these two triangles have 2 pairs of corresponding sides that are in proportion. (After all, as long as [br]a > 0 & b > 0, ka/a = k and kb/b = k, right?  )  [br][br]The [b]BLACK ANGLES INCLUDED[/b] between these two sides [b]ARE CONGRUENT[/b] as well.  [/color][br][br][b][color=#0000ff]From your observations, what can you conclude about the two triangles?  Why can you conclude this?[br]Clearly justify your response!  [/color][/b]
– Tim Brzezinski

4.2.

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4.3.

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4.4.

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4.5.

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4.6.

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4.7.

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4.8.

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4.9.

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4.10.

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4.11.

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4.12.

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4.13.

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4.14.

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4.15.

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4.16.

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4.17.

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4.18.

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4.19.

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4.20.

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4.21.

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4.22.

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4.23.

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4.24.

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4.25.

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4.26.

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4.27.

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5.

HSG.SRT.C.6 & C.7 & C.8

  • 1. Practice: Pythagorean Theorem (1)
  • 2. Quiz: Pythagorean Theorem (2B)
  • 3. Angles in Standard Position
  • 4. Animation 176
  • 5. Does R Matter?
  • 6. Quiz: Evaluating Trigonometric Functions of Angles Given a Point on its Terminal Ray
  • 7. Evaluating Trig Functions Given a Point on the Terminal Ray
  • 8. Trig Ratio Possibilities (Hints)
  • 9. Right Triangle Trig: Solving for Sides (2)
  • 10. Quiz: Special Right Triangles
  • 11. Special Angles in Standard Position: Intro (V2)
  • 12. Special Angles in Standard Position: Intro (V3)
  • 13. Special Angles in Standard Position: Intro (V1)
  • 14. Animation 177
  • 15. Animation 178
  • 16. Right Triangle Trigonometry: Intro
  • 17. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
  • 18. Trig Ratio Estimations (Right Triangle Context)
  • 19. Trig Ratio Estimations (Quiz: V1)
  • 20. Trig Ratio Estimations (Quiz: V2)
  • 21. Trigonometric Ratios (Right Triangle Context)
  • 22. Discovering Sine, Cosine, and Tangent in Right Triangles
  • 23. Special Trigonometric Identities: Quick Exploration
  • 24. What "CO" in COsine Means
  • 25. Quiz: Expressing Angles Using Inverse Trigonometric Function
  • 26. Animation 98
  • 27. Animation 99
  • 28. Animation 100
  • 29. YOUR Linear Speed?
  • 30. How Fast are You Spinning?

5.1.

Practice: Pythagorean Theorem (1)

– Tim Brzezinski

5.2.

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5.3.

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5.4.

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5.5.

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5.6.

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5.7.

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5.8.

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5.9.

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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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5.18.

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5.19.

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5.20.

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5.21.

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5.22.

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5.23.

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5.24.

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5.25.

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5.26.

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5.27.

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5.28.

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5.29.

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5.30.

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6.

HSG.SRT.D.9 & D.10 & D.11

  • 1. Triangle Investigation
  • 2. Law of Sines (& Area)
  • 3. Law of Sines (Illustrated)
  • 4. Law of Cosines: Discovery
  • 5. Bearing Illustrator (1)
  • 6. Direction Angle Illustrator
  • 7. Bearing Illustrator (2)
  • 8. Bearings Summary
  • 9. Bearing: Dynamic Illustration
  • 10. Bearings: Dynamic Illustrator

6.1.

Triangle Investigation

[b]Students:[/b][br][br]Please use this applet in conjunction with the activity you received at the beginning of today's class.
– Tim Brzezinski

6.2.

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6.3.

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6.4.

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6.5.

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6.6.

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6.7.

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6.8.

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6.9.

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6.10.

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7.

Links to Other CCSS High School: Geometry Standards

  • 1. Geometry Resources

7.1.

Geometry Resources

[list][*][b][url=https://www.geogebra.org/m/z8nvD94T]Congruence (Volume 1)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/munhXmzx]Congruence (Volume 2)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/dPqv8ACE]Similarity, Right Triangles, Trigonometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/C7dutQHh]Circles[/url][/b][/*][*][b][url=https://www.geogebra.org/m/K2YbdFk8]Coordinate and Analytic Geometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/xDNjSjEK]Area, Surface Area, Volume, 3D, Cross Section[/url] [/b][/*][*][b][url=https://www.geogebra.org/m/NjmEPs3t]Proof Challenges[/url]  [/b][/*][/list]
What phenomenon is dynamically being illustrated here? (Vertices are moveable.)
What phenomenon is dynamically being illustrated here? (Vertices are moveable.)
– Tim Brzezinski

8.

Additional - Links to CCSS High School: Functions Resources

  • 1. Functions Resources

8.1.

Functions Resources

[list][*][b][url=https://www.geogebra.org/m/k6Dvu9f3]Interpreting Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/uTddJKRC]Building Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/GMvvpwrm]Linear, Quadratic, and Exponential Functions[/url][/b][/*][*][b][url=https://www.geogebra.org/m/aWuJMDas]Trigonometric Functions[/url][/b][/*][/list]
Half-life function: Quick Exploration. (Large point & slider moveable.)
What does it mean for a function to be odd? (Points moveable.)
– GeoGebra Team, Tim Brzezinski

Information