HS Geometry

Tim Brzezinski, Aaron Dankman, Oct 31, 2016

Interactions for learning introductory geometry transformationally. Thank you to Tim Brzezinski for publishing the original material for this workbook.

Table of Contents

  1. Congruence and Transformations
    1. Proving Angles Congruent (3)
    2. Midpoint Definition (I)
    3. Animation 56
    4. Segment Bisectors (Definitions)
    5. Perpendicular Bisector Definition
    6. Animation 15
    7. Angle Bisector Definition (I)
    8. Complementary Meaning?
    9. Animation 48
    10. Supplementary Angles (Quick Exploration)
    11. Animation 19
    12. Circle vs. Sphere
    13. Animation 119
    14. Congruent Segments
  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?
  6. Links to Other CCSS High School: Geometry Standards
    1. Geometry Resources

1.

Congruence and Transformations

  • 1. Proving Angles Congruent (3)
  • 2. Midpoint Definition (I)
  • 3. Animation 56
  • 4. Segment Bisectors (Definitions)
  • 5. Perpendicular Bisector Definition
  • 6. Animation 15
  • 7. Angle Bisector Definition (I)
  • 8. Complementary Meaning?
  • 9. Animation 48
  • 10. Supplementary Angles (Quick Exploration)
  • 11. Animation 19
  • 12. Circle vs. Sphere
  • 13. Animation 119
  • 14. Congruent Segments

1.1.

Proving Angles Congruent (3)

In the applet below, use the provided transformational geometry tools within the limited toolbar [b]to prove [/b]that [color=#ff00ff][b]angle A [/b][/color][b]IS CONGRUENT TO [/b][color=#444444][b]angle B[/b][/color]. [br][br][color=#ff0000][b][u]Remember:[/u] [br][/b][/color][b]Using the Angle tool to simply find display the measures of angle A and angle B[/b][color=#ff0000] [b]IS NOT how we prove these 2 angles are congruent.[/b] [/color](The fact that congruent angles have equal measures is the [color=#ff0000][b]EFFECT[/b][/color] of these angles being congruent, [color=#ff0000][b]NOT the CAUSE.[/b][/color]) [br][br]If you don't recall what it means for angles to be considered congruent, [url=https://www.geogebra.org/m/evXSUG9J]click here[/url] for a refresher.
– Aaron Dankman, 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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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

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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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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5.3.

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5.4.

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5.5.

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5.6.

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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