CCSS High School: Geometry (Congruence) Volume 1

Tim Brzezinski, Bart Landenberger, Oct 1, 2016

[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. Congruent Angles: Definition
    4. Animation 36
    5. Midpoint Definition
    6. Animation 56
    7. Segment Bisectors (Definitions)
    8. Perpendicular Bisector Definition
    9. Animation 15
    10. Angle Bisector Definition (I)
    11. Complementary Meaning?
    12. Animation 48
    13. Supplementary Angles (Quick Exploration)
    14. Animation 19
    15. Circle vs. Sphere
  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. Reflections and Translations
    11. Tessellation by Translation
    12. Parallel Translation
    13. Transformations: Exercise 1
    14. Transformations: Exercise 2
    15. 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. Proving Tri's Congruent (I)
    2. Proving Tri's Congruent (II)
    3. SAS: Exercise 1
    4. SAS: Exercise 2
    5. SAS: Exercise 3
    6. SSS: Exercise 1
    7. SSS: Exercise 2
    8. SSS: Exercise 3
  5. HSG.CO.B.8
    1. SAS: Dynamic Proof!
    2. SSS: Dynamic Proof!
    3. SSS: Dynamically Illustrated
    4. ASA Theorem?
    5. Exploring SSA
    6. Is "SSA" Legit? What Do You Think?

1.

HSG.CO.A.1

  • 1. Congruent Segments: Quick Exploration
  • 2. Animation 18
  • 3. Congruent Angles: Definition
  • 4. Animation 36
  • 5. Midpoint Definition
  • 6. Animation 56
  • 7. Segment Bisectors (Definitions)
  • 8. Perpendicular Bisector Definition
  • 9. Animation 15
  • 10. Angle Bisector Definition (I)
  • 11. Complementary Meaning?
  • 12. Animation 48
  • 13. Supplementary Angles (Quick Exploration)
  • 14. Animation 19
  • 15. Circle vs. Sphere

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

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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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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1.13.

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1.14.

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1.15.

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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. Reflections and Translations
  • 11. Tessellation by Translation
  • 12. Parallel Translation
  • 13. Transformations: Exercise 1
  • 14. Transformations: Exercise 2
  • 15. 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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2.3.

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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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2.9.

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2.10.

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2.11.

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2.12.

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2.13.

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2.14.

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2.15.

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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

3.2.

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4.

HSG.CO.B.7

  • 1. Proving Tri's Congruent (I)
  • 2. Proving Tri's Congruent (II)
  • 3. SAS: Exercise 1
  • 4. SAS: Exercise 2
  • 5. SAS: Exercise 3
  • 6. SSS: Exercise 1
  • 7. SSS: Exercise 2
  • 8. SSS: Exercise 3

4.1.

Proving Tri's Congruent (I)

[color=#000000]Recall an [b]isometry[/b] is a [b]transformation that preserves distance. [br][/b][br]Also recall that, by definition, 2 polygons are said to be congruent polygons if and only if one polygon can be mapped perfectly onto the other polygon using an isometry or a composition of two or more isometries. [br][br]Use the tools of GeoGebra to show, by definition, that the following two triangles are congruent. [/color]
– 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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5.

HSG.CO.B.8

  • 1. SAS: Dynamic Proof!
  • 2. SSS: Dynamic Proof!
  • 3. SSS: Dynamically Illustrated
  • 4. ASA Theorem?
  • 5. Exploring SSA
  • 6. Is "SSA" Legit? What Do You Think?

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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