CCSS High School: Geometry (Congruence) Volume 1

Tim Brzezinski, Jonathan Talley, May 5, 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.CO.A.1
    1. Congruent Segments: Quick Exploration
    2. Animation 18
    3. Proving Segments Congruent (Exercise)
    4. Congruent Angles: Definition
    5. Animation 36
    6. Proving Angles Congruent (1)
    7. Proving Angles Congruent (2)
    8. Proving Angles Congruent
    9. Midpoint Definition
    10. Animation 56
    11. Segment Bisector Definition
    12. Animation 160
    13. Perpendicular Lines: Quick Intro
    14. Segment Bisectors (Definitions)
    15. Perpendicular Bisector Definition
    16. Perpendicular Bisector Definition
    17. Animation 15
    18. Angle Bisector Definition (I)
    19. Complementary Meaning?
    20. Animation 48
    21. Supplementary Angles (Quick Exploration)
    22. Animation 19
    23. Circle vs. Sphere
    24. Triangle Altitude Illustrator and Definition Writing Prompt
    25. Animation 162
    26. Triangle Median Illustrator & Definition Writing Prompt
    27. Animation 161
    28. Equilateral Action!!!
    29. Animation 165
    30. Equiangular Action!!!
    31. Animation 166
    32. Regular Polygon Action!
    33. Animation 193
  2. HSG.CO.A.2, A.3, A.4, A.5
    1. Translations and Rotations
    2. Translating Triangle by a Vector
    3. Reflections
    4. Reflections Introduction
    5. Flip Flop Reflection
    6. Tool for compositions of rigid transformations and dilations
    7. Applet for Multiple Transformations
    8. Messing With Lisa
    9. Rotations: Introduction
    10. Point Symmetry: Another Perspective
    11. Reflections and Translations
    12. Tessellation by Translation
    13. Parallel Translation
    14. Transformations: Exercise 1
    15. Transformations: Exercise 2
    16. Transformations: Exercise 3
  3. HSG.CO.B.6
    1. Are the triangles congruent (part 2)?
    2. Are the triangles congruent (part 3)?
  4. HSG.CO.B.7
    1. Congruent Figures: Dynamic Illustration
    2. Animation 242
    3. Proving Tri's Congruent (I)
    4. Proving Tri's Congruent (II)
    5. SAS - Exercise 1A
    6. SAS - Exercise 1B
    7. SAS - Exercise 2
    8. SAS - Exercise 3
    9. SSS: Exercise 1
    10. SSS: Exercise 2
    11. SSS: Exercise 3
  5. HSG.CO.B.8
    1. SAS: Dynamic Proof!
    2. Animation 136
    3. SSS: Dynamic Proof!
    4. SSS: Dynamically Illustrated
    5. Animation 138
    6. ASA Theorem?
    7. Animation 137
    8. SSA Theorem?
    9. Is "SSA" Legit? What Do You Think?
    10. Exploring SSA
    11. HL: Hypotenuse-Leg Action!
    12. Animation 225
  6. Links to Other CCSS High School: Geometry Standards
    1. Geometry Resources

1.

HSG.CO.A.1

  • 1. Congruent Segments: Quick Exploration
  • 2. Animation 18
  • 3. Proving Segments Congruent (Exercise)
  • 4. Congruent Angles: Definition
  • 5. Animation 36
  • 6. Proving Angles Congruent (1)
  • 7. Proving Angles Congruent (2)
  • 8. Proving Angles Congruent
  • 9. Midpoint Definition
  • 10. Animation 56
  • 11. Segment Bisector Definition
  • 12. Animation 160
  • 13. Perpendicular Lines: Quick Intro
  • 14. Segment Bisectors (Definitions)
  • 15. Perpendicular Bisector Definition
  • 16. Perpendicular Bisector Definition
  • 17. Animation 15
  • 18. Angle Bisector Definition (I)
  • 19. Complementary Meaning?
  • 20. Animation 48
  • 21. Supplementary Angles (Quick Exploration)
  • 22. Animation 19
  • 23. Circle vs. Sphere
  • 24. Triangle Altitude Illustrator and Definition Writing Prompt
  • 25. Animation 162
  • 26. Triangle Median Illustrator & Definition Writing Prompt
  • 27. Animation 161
  • 28. Equilateral Action!!!
  • 29. Animation 165
  • 30. Equiangular Action!!!
  • 31. Animation 166
  • 32. Regular Polygon Action!
  • 33. Animation 193

1.1.

Congruent Segments: Quick Exploration

[color=#000000]The following app illustrates what it means for segments to be classified as [b]congruent segments.  [br][/b][/color]
Interact with the applet below for a few minutes. Be sure to move the LARGE POINTS around each time before you drag the slider! Then answer the questions that follow.
What transformations have we learned about so far? List them.
What transformations did you observe here while interacting with this app?
What does it mean for segments to be classified as [b]congruent segments[/b]? Explain.
– Tim Brzezinski

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1.11.

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1.13.

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1.15.

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1.17.

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1.21.

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1.25.

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1.26.

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1.27.

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1.28.

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1.29.

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1.30.

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1.31.

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1.32.

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1.33.

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2.

HSG.CO.A.2, A.3, A.4, A.5

  • 1. Translations and Rotations
  • 2. Translating Triangle by a Vector
  • 3. Reflections
  • 4. Reflections Introduction
  • 5. Flip Flop Reflection
  • 6. Tool for compositions of rigid transformations and dilations
  • 7. Applet for Multiple Transformations
  • 8. Messing With Lisa
  • 9. Rotations: Introduction
  • 10. Point Symmetry: Another Perspective
  • 11. Reflections and Translations
  • 12. Tessellation by Translation
  • 13. Parallel Translation
  • 14. Transformations: Exercise 1
  • 15. Transformations: Exercise 2
  • 16. Transformations: Exercise 3

2.1.

Translations and Rotations

Direct Isometries
Translations and rotations are [i]direct isometries[/i]. If you read the names of the vertices in cyclic order (A-B-C and A'-B'-C'), both would be read in the counterclockwise (or clockwise) direction.
Translation
Rotation
– Steve Phelps

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2.11.

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2.12.

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2.14.

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3.

HSG.CO.B.6

  • 1. Are the triangles congruent (part 2)?
  • 2. Are the triangles congruent (part 3)?

3.1.

Are the triangles congruent (part 2)?

Use the given measurement tools to establish that the corresponding sides of triangle ABS and [br]triangle A'B'C' are parallel and have equal lengths.[br][br]Use the given transformation tools to establish that triangle ABC is congruent to triangle A'B'C'.
– Steve Phelps

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4.

HSG.CO.B.7

  • 1. Congruent Figures: Dynamic Illustration
  • 2. Animation 242
  • 3. Proving Tri's Congruent (I)
  • 4. Proving Tri's Congruent (II)
  • 5. SAS - Exercise 1A
  • 6. SAS - Exercise 1B
  • 7. SAS - Exercise 2
  • 8. SAS - Exercise 3
  • 9. SSS: Exercise 1
  • 10. SSS: Exercise 2
  • 11. SSS: Exercise 3

4.1.

Congruent Figures: Dynamic Illustration

[color=#0000ff]Recall an ISOMETRY is a transformation that preserves distance.[/color] So far, we have already explored the following isometries:[br][br][color=#0000ff]Translation by Vector[br]Rotation about a Point[br]Reflection about a Line[br]Reflection about a Point (same as 180-degree rotation about a point) [/color][br][br]For a quick refresher about [color=#0000ff]isometries[/color], see this [url=https://www.geogebra.org/m/KFtdRvyv]Messing with Mona applet[/url].
CONGRUENT FIGURES
[b]Definition: [br][br]Any two figures are said to be CONGRUENT if and only if one can be mapped perfectly onto the other using [color=#0000ff]any 1 or composition of 2 (or more) ISOMETRIES.[/color][/b][br][br]The applet below dynamically illustrates, [b]by DEFINITION[/b], what it means for any 2 figures (in this case, triangles) to be [b]CONGRUENT.[/b] [br][br]Feel free to move the BIG WHITE VERTICES of either triangle anywhere you'd like at any time.
Quick (Silent) Demo
– Tim Brzezinski

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5.

HSG.CO.B.8

  • 1. SAS: Dynamic Proof!
  • 2. Animation 136
  • 3. SSS: Dynamic Proof!
  • 4. SSS: Dynamically Illustrated
  • 5. Animation 138
  • 6. ASA Theorem?
  • 7. Animation 137
  • 8. SSA Theorem?
  • 9. Is "SSA" Legit? What Do You Think?
  • 10. Exploring SSA
  • 11. HL: Hypotenuse-Leg Action!
  • 12. Animation 225

5.1.

SAS: Dynamic Proof!

[color=#000000]The [/color][b][u][color=#0000ff]SAS Triangle Congruence Theorem[/color][/u][/b][color=#000000] states that [/color][b][color=#000000]if 2 sides [/color][color=#000000]and their [/color][color=#ff00ff]included angle [/color][color=#000000]of one triangle are congruent to 2 sides and their [/color][color=#ff00ff]included angle [/color][color=#000000]of another triangle, then those triangles are congruent. [/color][/b][color=#000000]The applet below uses transformational geometry to dynamically prove this very theorem. [br][br][/color][color=#000000]Interact with this applet below for a few minutes, then answer the questions that follow. [br][/color][color=#000000]As you do, feel free to move the [b]BIG WHITE POINTS[/b] anywhere you'd like on the screen! [/color]
Q1:
What geometry transformations did you observe in the applet above? List them.
Q2:
What common trait do all these transformations (you listed in your response to (1)) have?
Q3:
Go to [url=https://www.geogebra.org/m/d9HrmyAp#chapter/74321]this link[/url] and complete the first 5 exercises in this GeoGebra Book chapter.
Quick (Silent) Demo
– Tim Brzezinski

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6.

Links to Other CCSS High School: Geometry Standards

  • 1. Geometry Resources

6.1.

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