CCSS High School: Geometry (Similarity, Right T, Trig.)

Tim Brzezinski, Israel, Apr 20, 2017

[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. Dilating a Point (Intro)
    3. Dilating a Line: HSG.SRT.A.1.A
    4. Dilating a Segment: HSG.SRT.A.1.B
  2. HSG.SRT.A.2
    1. Similar Figures: Dynamic Illustration
    2. Properties of Dilations
    3. Properties of Dilations (II)
    4. Proving Triangles Similar (1)
    5. Proving Triangles Similar (2)
    6. Proving Triangles Similar (3)
    7. 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. Animation 13
    8. Pythagorean and similar triangles.
    9. Effortless Trisections
    10. Animation 134
    11. Parallel Lines Congruent Segments Theorem
    12. Parallel Lines Proportionality Theorem
    13. Triangle-Angle Bisector Theorem
    14. Animation 81
    15. Proof Without Words
    16. Wordless Pythagorean Theorem based on Translations
    17. Proof Without Words
    18. Proof Without Words
    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
  5. HSG.SRT.C.6 & C.7 & C.8
    1. Angles in Standard Position
    2. Animation 176
    3. Special Angles in Standard Position: Intro (V2)
    4. Special Angles in Standard Position: Intro (V3)
    5. Special Angles in Standard Position: Intro (V1)
    6. Animation 177
    7. Animation 178
    8. Right Triangle Trigonometry: Intro
    9. Discovering Sine, Cosine, and Tangent in Right Triangles
    10. Trigonometric Ratios (Right Triangle Context)
    11. What "CO" in COsine Means
    12. Animation 98
    13. Animation 99
    14. Animation 100
    15. How Fast are You Spinning?
  6. HSG.SRT.D.9 & D.10 & D.11
    1. Law of Sines (& Area)
    2. Law of Sines (Illustrated)
    3. Law of Cosines: Discovery
  7. Links to Other CCSS High School: Geometry Standards
    1. Geometry Resources

1.

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

  • 1. Dilation Introduction: Transformers!
  • 2. Dilating a Point (Intro)
  • 3. Dilating a Line: HSG.SRT.A.1.A
  • 4. 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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2.

HSG.SRT.A.2

  • 1. Similar Figures: Dynamic Illustration
  • 2. Properties of Dilations
  • 3. Properties of Dilations (II)
  • 4. Proving Triangles Similar (1)
  • 5. Proving Triangles Similar (2)
  • 6. Proving Triangles Similar (3)
  • 7. 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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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. Animation 13
  • 8. Pythagorean and similar triangles.
  • 9. Effortless Trisections
  • 10. Animation 134
  • 11. Parallel Lines Congruent Segments Theorem
  • 12. Parallel Lines Proportionality Theorem
  • 13. Triangle-Angle Bisector Theorem
  • 14. Animation 81
  • 15. Proof Without Words
  • 16. Wordless Pythagorean Theorem based on Translations
  • 17. Proof Without Words
  • 18. Proof Without Words
  • 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

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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5.

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

  • 1. Angles in Standard Position
  • 2. Animation 176
  • 3. Special Angles in Standard Position: Intro (V2)
  • 4. Special Angles in Standard Position: Intro (V3)
  • 5. Special Angles in Standard Position: Intro (V1)
  • 6. Animation 177
  • 7. Animation 178
  • 8. Right Triangle Trigonometry: Intro
  • 9. Discovering Sine, Cosine, and Tangent in Right Triangles
  • 10. Trigonometric Ratios (Right Triangle Context)
  • 11. What "CO" in COsine Means
  • 12. Animation 98
  • 13. Animation 99
  • 14. Animation 100
  • 15. How Fast are You Spinning?

5.1.

Angles in Standard Position

The angle drawn below in the coordinate plane is classified as being drawn in [b]STANDARD POSITION. [br][br][/b]Interact with the applet for a minute.[br]Then answer the question that follows.
ANGLE IN STANDARD POSITION:
1.
What does it mean for an angle drawn in the coordinate plane to be drawn in [b]STANDARD POSITION? [br][br][/b](Your definition should list 2 criteria.)
– 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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6.

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

  • 1. Law of Sines (& Area)
  • 2. Law of Sines (Illustrated)
  • 3. Law of Cosines: Discovery

6.1.

Law of Sines (& Area)

Interact with the applet below for a minute. [br]Then, answer the questions that follow. [br](Please don't slide the 2nd slider until prompted to in the directions below.)
1) Take a look at the yellow right triangle on the left.[br]Write an equation that expresses the relationship among angle [i]B[/i], the triangle's height, and side [i]c[/i].
2) Rewrite this equation so that [i]height [/i]is written in terms of side [i]c[/i] and angle [i]B[/i].
3) Now consider the pink right triangle on the right. Write an equation that expresses the relationship among angle [i]C[/i], side [i]b[/i], and the triangle's height. [br]
4) Rewrite this equation so that [i]height[/i] is written in terms of side [i]b[/i] and angle [i]C[/i].
5) Take your responses to questions (2) and (4) to write a new equation that expresses the relationship among [i]C[/i], [i]B[/i], [i]c[/i], and [i]b[/i]. Write this equation so that [i]C[/i] and [i]c[/i] appear on one side of the equation and that [i]B[/i] and [i]b[/i] appear on the other.
6) Now drag the slider in the upper right hand corner. Now, given the fact that the length of segment [i]BC[/i] would be denoted as [i]a [/i](it's just not drawn in the applet above), write an expression for the area of this original triangle in terms of [i]a[/i], [i]b[/i], and [i]C[/i].
7) Same question as in (6) above, but this time write the area of the triangle in terms of [i]a[/i], [i]c[/i], and [i]B[/i].
8) Suppose that dragging the first slider dropped a height from point [i]C[/i] instead of point [i]A[/i]. Answer questions (1) - (5) again, this time letting [i]c[/i] serve as the base of this triangle (vs. side [i]a[/i]). Notice anything interesting in your results?
– Tim Brzezinski

6.2.

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6.3.

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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

Information