CCSS High School: Geometry (Congruence) Volume 2

Tim Brzezinski, Mr. Mundey, Nov 13, 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: Steve Phelps ([url]https://www.geogebra.org/stevephelps[/url]) Jennifer Silverman ([url]https://www.geogebra.org/jennifer+silverman[/url]) Without their amazing talent and creativity, this resource would not have been possible. [/b][/color]

Table of Contents

  1. HSG.CO.C.9
    1. Proof Prompt (A)
    2. Animation 128
    3. Proof Prompt (B)
    4. Animation 127
    5. Proof Prompt (C)
    6. Animation 131
    7. Proof Prompt (D)
    8. Animation 130
    9. Vertical Angles Exploration (1)
    10. Vertical Angles Exploration (2)
    11. Vertical Angles Theorem
    12. Animation 1
    13. Transversal Intersects Parallel Lines
    14. Corresponding Angles: Quick Investigation
    15. Congruent Corresponding Angles to Start? (Quick Investigation)
    16. Exploring Corresponding Angles (V2)
    17. Alternate Interior Angles: Quick Investigation
    18. Alternate Interior Angles Theorem (V1)
    19. Exploring Alternate Interior Angles (V2)
    20. Alternate Interior Angles Theorem (V3)
    21. Animation 16
    22. Alternate Exterior Angles: Quick Investigation
    23. Alternate Exterior Angles Theorem (V1)
    24. Exploring Alternate Exterior Angles (V2)
    25. Animation 39
    26. Animation 60
    27. Same-Side-Interior Angles: Quick Investigation
    28. Same Side Interior Angles Theorem
    29. Animation 29
    30. Same-Side-Exterior Angles: Quick Investigation
    31. Exploring Same Side Exterior Angles
    32. Animation 73
    33. Solution to Parallel Lines & Angles Problem
    34. A Special Theorem: Part 1 (V1)
    35. A Special Theorem: Part 2 (V1)
    36. Animation 57
    37. Another Special Theorem: Part 1 (V1)
    38. Another Special Theorem: Part 2 (V1)
    39. Another Special Theorem: Part 1 (V2)
    40. Another Special Theorem: Part 1 (V3)
    41. Animation 58
    42. Effortless Trisections
    43. Animation 134
    44. Parallel Lines Congruent Segments Theorem
    45. Parallel Lines Proportionality Theorem
    46. Animation 135
  2. HSG.CO.C.10
    1. Area of a Triangle (Discovery)
    2. Triangle Angle Theorems
    3. Triangle Angle Theorems (V2)
    4. Triangle Angle Theorems (V3)
    5. Transformation Proof 3
    6. Triangle Angle Sum Theorem
    7. Animation 2
    8. Obvious Corollary
    9. Animation 140
    10. Triangle Exterior Angle
    11. Animation 102
    12. Angles of a Right Triangle
    13. Animation 104
    14. Acute Angles of a Right Triangle
    15. Animation 129
    16. Exterior Angles of a Triangle
    17. Animation 22
    18. Isosceles Triangle Theorem: Discovery Lab
    19. A Special Triangle & Its Properties (I)
    20. Animation 8
    21. Converse of IST (V1)
    22. Animation 82
    23. Animation 23
    24. Another Special Triangle and its Properties (II)
    25. Animation 27
    26. Converse of IST (II)
    27. Animation 84
    28. Special Right Triangle (I)
    29. Animation 59
    30. Special Right Triangle (II)
    31. Another Special Right Triangle (Guided Discovery)
    32. Animation 42
    33. Triangle Midsegment Action!
    34. Midsegment of a Triangle
    35. Special Line through Triangle V1 (Theorem Discovery)
    36. Points of Concurrency: Investigation
    37. Orthocenter Exploration
    38. Circumcenter (& Questions)
    39. Circumcenter & Circumcircle Action!
    40. Animation 28
    41. Midpoint of HYP
    42. Animation 46
    43. Incenter Exploration (A)
    44. Incenter Exploration (B)
    45. Incenter & Incircle Action!
    46. Animation 17
    47. Medians Centroid Theorem (Proof without Words)
    48. Medians and Centroid Dance
    49. Animation 25
    50. Medians & Equal Areas!
    51. Animation 74
    52. Animation 90
    53. Euler Line (Informal Investigation)
    54. Euler Line Generator
    55. Morley Action!
    56. Animation 113
    57. Napoleon Motion!
    58. Animation 14
    59. Vivani Action!
    60. Equilateral Action (1)!
    61. Animation 91
    62. Equilateral Action (2)!
    63. Animation 94
    64. Equilateral Action (3)!
    65. Animation 107
    66. Equilateral Action (4)!
    67. Animation 108
    68. Triangle Inequality
    69. Triangle Inequality (II)
    70. NCTM Calendar Problem (11-2-2016)
    71. Animation 139
    72. 9 Point Circle Action
    73. Animation 34
    74. 9-Point Circle Action (Part 2)
    75. Animation 106
    76. 9-Point Circle Action (Part 3A)
    77. Animation 71
    78. 9-Point Circle Action (Part 3B)
    79. 9-Point Circle Action (Part 4)
    80. Animation 53
    81. 9-Point Circle Action (Part 5)
    82. Animation 115
    83. Isosceles Triangle: Surprising Theorem!
    84. Animation 121
    85. 9-Point Circle (Informal Investigation)
  3. HSG.CO.C.11
    1. Parallelogram: Area
    2. Parallelogram Construction Template & Discovery Lab
    3. Parallelogram Template (Scaffolded Discovery)
    4. Parallelogram Template
    5. Parallelogram: Theorem 1
    6. Animation 123
    7. Parallelogram: Theorem 2
    8. Animation 124
    9. Parallelogram: Theorem (3)
    10. Animation 125
    11. Parallelograms (I)
    12. Animation 75
    13. Animation 80
    14. Animation 6
    15. What Kind of Quadrilateral is This? (Part 1)
    16. Congruent Opposite Sides of a Quadrilateral
    17. Quadrilateral With Opposite Angles Congruent
    18. What Kind of Quad?
    19. Quadrilateral Investigation (I)
    20. Rhombus Action!
    21. Animation 11
    22. Rhombus Template (Scaffolded Discovery)
    23. Rhombus Template with Investigation Questions
    24. Rectangle Action!
    25. Animation 31
    26. Rectangle Template (Scaffolded Investigation)
    27. Rectangle Template with Investigation Questions
    28. Square Action!
    29. Animation 41
    30. Square Template (Scaffolded Investigation)
    31. Square Template with Investigation Questions
    32. Trapezoids: Median Investiagtion
    33. Trapezoid Median (Midsegment) Action!
    34. Animation 67
    35. Trapezoid Median: 2 Discoveries
    36. Isosceles Trapezoid Action!
    37. Animation 64
    38. Animation 35
    39. Isosceles Trapezoid Template (Scaffolded Discovery)
    40. Isosceles Trapezoid Template
    41. Animation 49
    42. Kite Action!
    43. Animation 26
    44. Kite Template (Scaffolded Investigation)
    45. Quad Midpoints Action!
    46. Animation 52
    47. Thebault Action!!!
    48. Animation 83
    49. Finsler-Hadwiger Action!!!
    50. Animation 110
    51. Van Aubel Action!!!
    52. Animation 109
    53. Square Problem
    54. 3 Squares Problem
    55. Animation 120
  4. HSG.CO.D.12
    1. Congruent Segments Construction (I)
    2. Congruent Angle: Construction
    3. Alternate Angle Bisector Construction
    4. Animation 117
    5. Angle Bisector: Construction
    6. Perpendicular Line: Construction 1
    7. Perpendicular Line: Construction 2
    8. Perpendicular Bisector: Construction
    9. Parallel Line: Constructions
    10. Congruent Triangle: Construction
    11. Other Constructions: 1
    12. Other Constructions: 2
    13. Centroid Construction Challenge #1
    14. Possible Solution for Centroid Challenge #1
    15. Centroid Construction Challenge #2
    16. Possible Solution for Centroid Construction Challenge #2
    17. Centroid Construction Challenge #3
    18. Possible solution to Centroid Construction Challenge #3
    19. Triangle Construction Challenge #1
    20. Possible solution to Triangle Construction Challenge #1
    21. Triangle Consruction Challenge #2
    22. Possible solution to Triangle Construction Challenge #2
    23. Triangle Construction Challenge #3
    24. Potential solution for Triangle Construction Challenge #3
    25. Bizzare Trisection?
  5. HSG.CO.D.13
    1. Equilateral Triangle Construction Template
    2. Square Construction Template
    3. Regular Hexagon Construction Template
  6. Links to Other CCSS High School: Geometry Standards
    1. Geometry Resources

1.

HSG.CO.C.9

  • 1. Proof Prompt (A)
  • 2. Animation 128
  • 3. Proof Prompt (B)
  • 4. Animation 127
  • 5. Proof Prompt (C)
  • 6. Animation 131
  • 7. Proof Prompt (D)
  • 8. Animation 130
  • 9. Vertical Angles Exploration (1)
  • 10. Vertical Angles Exploration (2)
  • 11. Vertical Angles Theorem
  • 12. Animation 1
  • 13. Transversal Intersects Parallel Lines
  • 14. Corresponding Angles: Quick Investigation
  • 15. Congruent Corresponding Angles to Start? (Quick Investigation)
  • 16. Exploring Corresponding Angles (V2)
  • 17. Alternate Interior Angles: Quick Investigation
  • 18. Alternate Interior Angles Theorem (V1)
  • 19. Exploring Alternate Interior Angles (V2)
  • 20. Alternate Interior Angles Theorem (V3)
  • 21. Animation 16
  • 22. Alternate Exterior Angles: Quick Investigation
  • 23. Alternate Exterior Angles Theorem (V1)
  • 24. Exploring Alternate Exterior Angles (V2)
  • 25. Animation 39
  • 26. Animation 60
  • 27. Same-Side-Interior Angles: Quick Investigation
  • 28. Same Side Interior Angles Theorem
  • 29. Animation 29
  • 30. Same-Side-Exterior Angles: Quick Investigation
  • 31. Exploring Same Side Exterior Angles
  • 32. Animation 73
  • 33. Solution to Parallel Lines & Angles Problem
  • 34. A Special Theorem: Part 1 (V1)
  • 35. A Special Theorem: Part 2 (V1)
  • 36. Animation 57
  • 37. Another Special Theorem: Part 1 (V1)
  • 38. Another Special Theorem: Part 2 (V1)
  • 39. Another Special Theorem: Part 1 (V2)
  • 40. Another Special Theorem: Part 1 (V3)
  • 41. Animation 58
  • 42. Effortless Trisections
  • 43. Animation 134
  • 44. Parallel Lines Congruent Segments Theorem
  • 45. Parallel Lines Proportionality Theorem
  • 46. Animation 135

1.1.

Proof Prompt (A)

[color=#0000ff][b]Theorem: [/b][/color][br][br]If two lines cross to form [color=#9900ff][b]congruent, adjacent angles[/b][/color], then those [b]lines are perpendicular.[/b] [br][br][br]Write a 2-column (or paragraph) proof that formally proves what this applet informally illustrates. [br][i][color=#cc0000]Be sure to set up an appropriate diagram, state what is given, and state what needs to be proven. [br]Then prove it! [/color][/i] [br][br]
Proof Prompt (A)
– Tim Brzezinski

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2.

HSG.CO.C.10

  • 1. Area of a Triangle (Discovery)
  • 2. Triangle Angle Theorems
  • 3. Triangle Angle Theorems (V2)
  • 4. Triangle Angle Theorems (V3)
  • 5. Transformation Proof 3
  • 6. Triangle Angle Sum Theorem
  • 7. Animation 2
  • 8. Obvious Corollary
  • 9. Animation 140
  • 10. Triangle Exterior Angle
  • 11. Animation 102
  • 12. Angles of a Right Triangle
  • 13. Animation 104
  • 14. Acute Angles of a Right Triangle
  • 15. Animation 129
  • 16. Exterior Angles of a Triangle
  • 17. Animation 22
  • 18. Isosceles Triangle Theorem: Discovery Lab
  • 19. A Special Triangle & Its Properties (I)
  • 20. Animation 8
  • 21. Converse of IST (V1)
  • 22. Animation 82
  • 23. Animation 23
  • 24. Another Special Triangle and its Properties (II)
  • 25. Animation 27
  • 26. Converse of IST (II)
  • 27. Animation 84
  • 28. Special Right Triangle (I)
  • 29. Animation 59
  • 30. Special Right Triangle (II)
  • 31. Another Special Right Triangle (Guided Discovery)
  • 32. Animation 42
  • 33. Triangle Midsegment Action!
  • 34. Midsegment of a Triangle
  • 35. Special Line through Triangle V1 (Theorem Discovery)
  • 36. Points of Concurrency: Investigation
  • 37. Orthocenter Exploration
  • 38. Circumcenter (& Questions)
  • 39. Circumcenter & Circumcircle Action!
  • 40. Animation 28
  • 41. Midpoint of HYP
  • 42. Animation 46
  • 43. Incenter Exploration (A)
  • 44. Incenter Exploration (B)
  • 45. Incenter & Incircle Action!
  • 46. Animation 17
  • 47. Medians Centroid Theorem (Proof without Words)
  • 48. Medians and Centroid Dance
  • 49. Animation 25
  • 50. Medians & Equal Areas!
  • 51. Animation 74
  • 52. Animation 90
  • 53. Euler Line (Informal Investigation)
  • 54. Euler Line Generator
  • 55. Morley Action!
  • 56. Animation 113
  • 57. Napoleon Motion!
  • 58. Animation 14
  • 59. Vivani Action!
  • 60. Equilateral Action (1)!
  • 61. Animation 91
  • 62. Equilateral Action (2)!
  • 63. Animation 94
  • 64. Equilateral Action (3)!
  • 65. Animation 107
  • 66. Equilateral Action (4)!
  • 67. Animation 108
  • 68. Triangle Inequality
  • 69. Triangle Inequality (II)
  • 70. NCTM Calendar Problem (11-2-2016)
  • 71. Animation 139
  • 72. 9 Point Circle Action
  • 73. Animation 34
  • 74. 9-Point Circle Action (Part 2)
  • 75. Animation 106
  • 76. 9-Point Circle Action (Part 3A)
  • 77. Animation 71
  • 78. 9-Point Circle Action (Part 3B)
  • 79. 9-Point Circle Action (Part 4)
  • 80. Animation 53
  • 81. 9-Point Circle Action (Part 5)
  • 82. Animation 115
  • 83. Isosceles Triangle: Surprising Theorem!
  • 84. Animation 121
  • 85. 9-Point Circle (Informal Investigation)

2.1.

Area of a Triangle (Discovery)

[color=#000000]Interact with the applet below for a few minutes.  [br][br][i]Be sure to move the [/i][b]VERTICES[/b] [i]of the triangle around each time [b]before you move the slider. [br][/b][/i][/color][br]Answer the questions that appear below the applet.
1
What LARGER FIGURE was formed when the slider reached its end? How do we know this to be true?
2.
How does the area of the original triangle compare with the area of this LARGER FIGURE?
3.
How do we find the area of this LARGER FIGURE? What is the formula we use to find it?
4.
Given your responses to (2) & (3), write a formula that gives the area of JUST ONE of these congruent triangles.
Quick (Silent) Demo
– Tim Brzezinski

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3.

HSG.CO.C.11

  • 1. Parallelogram: Area
  • 2. Parallelogram Construction Template & Discovery Lab
  • 3. Parallelogram Template (Scaffolded Discovery)
  • 4. Parallelogram Template
  • 5. Parallelogram: Theorem 1
  • 6. Animation 123
  • 7. Parallelogram: Theorem 2
  • 8. Animation 124
  • 9. Parallelogram: Theorem (3)
  • 10. Animation 125
  • 11. Parallelograms (I)
  • 12. Animation 75
  • 13. Animation 80
  • 14. Animation 6
  • 15. What Kind of Quadrilateral is This? (Part 1)
  • 16. Congruent Opposite Sides of a Quadrilateral
  • 17. Quadrilateral With Opposite Angles Congruent
  • 18. What Kind of Quad?
  • 19. Quadrilateral Investigation (I)
  • 20. Rhombus Action!
  • 21. Animation 11
  • 22. Rhombus Template (Scaffolded Discovery)
  • 23. Rhombus Template with Investigation Questions
  • 24. Rectangle Action!
  • 25. Animation 31
  • 26. Rectangle Template (Scaffolded Investigation)
  • 27. Rectangle Template with Investigation Questions
  • 28. Square Action!
  • 29. Animation 41
  • 30. Square Template (Scaffolded Investigation)
  • 31. Square Template with Investigation Questions
  • 32. Trapezoids: Median Investiagtion
  • 33. Trapezoid Median (Midsegment) Action!
  • 34. Animation 67
  • 35. Trapezoid Median: 2 Discoveries
  • 36. Isosceles Trapezoid Action!
  • 37. Animation 64
  • 38. Animation 35
  • 39. Isosceles Trapezoid Template (Scaffolded Discovery)
  • 40. Isosceles Trapezoid Template
  • 41. Animation 49
  • 42. Kite Action!
  • 43. Animation 26
  • 44. Kite Template (Scaffolded Investigation)
  • 45. Quad Midpoints Action!
  • 46. Animation 52
  • 47. Thebault Action!!!
  • 48. Animation 83
  • 49. Finsler-Hadwiger Action!!!
  • 50. Animation 110
  • 51. Van Aubel Action!!!
  • 52. Animation 109
  • 53. Square Problem
  • 54. 3 Squares Problem
  • 55. Animation 120

3.1.

Parallelogram: Area

Directions:
In the app below, use the [b]filling[/b] slider to make the parallelogram light enough so that you can see the white gridlines through it. [b]But don't touch anything else yet! [/b][br][br]After you set the [b]filling[/b] slider, try to count the number of squares inside this parallelogram. [br]Be sure to include partial squares! Provide a good estimate in the question box below.
Try to count the number of squares inside this parallelogram. How many squares (i.e. square units) do you estimate to be inside this parallelogram?
Slide the "[b]Slide Me" [/b]slider now. Carefully observe what happens. What shape do you see now?
How does the area of this new shape compare with the area of the original parallelogram? How do you know this?
How many squares do you now count in the newer shape that was formed? How many squares were in the original parallelogram?
Without looking it up on another tab in your browser, describe how we can find the area of ANY PARALLELOGRAM.
– Tim Brzezinski

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4.

HSG.CO.D.12

  • 1. Congruent Segments Construction (I)
  • 2. Congruent Angle: Construction
  • 3. Alternate Angle Bisector Construction
  • 4. Animation 117
  • 5. Angle Bisector: Construction
  • 6. Perpendicular Line: Construction 1
  • 7. Perpendicular Line: Construction 2
  • 8. Perpendicular Bisector: Construction
  • 9. Parallel Line: Constructions
  • 10. Congruent Triangle: Construction
  • 11. Other Constructions: 1
  • 12. Other Constructions: 2
  • 13. Centroid Construction Challenge #1
  • 14. Possible Solution for Centroid Challenge #1
  • 15. Centroid Construction Challenge #2
  • 16. Possible Solution for Centroid Construction Challenge #2
  • 17. Centroid Construction Challenge #3
  • 18. Possible solution to Centroid Construction Challenge #3
  • 19. Triangle Construction Challenge #1
  • 20. Possible solution to Triangle Construction Challenge #1
  • 21. Triangle Consruction Challenge #2
  • 22. Possible solution to Triangle Construction Challenge #2
  • 23. Triangle Construction Challenge #3
  • 24. Potential solution for Triangle Construction Challenge #3
  • 25. Bizzare Trisection?

4.1.

Congruent Segments Construction (I)

I[color=#000000]n the applet below, use the limited toolbar to construct segments that are congruent to the two segments displayed.[br]Use the [b]Distance[/b] tool afterwards to show that these segments are indeed congruent. [/color]
I[color=#000000]n the applet below, use the limited toolbar to construct segments that are [b]triple the length[/b] of the two segments displayed.[br]Use the [b]Distance[/b] tool afterwards to show that these segments are indeed triple the length of the original segments displayed. [/color]
– Tim Brzezinski

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5.

HSG.CO.D.13

  • 1. Equilateral Triangle Construction Template
  • 2. Square Construction Template
  • 3. Regular Hexagon Construction Template

5.1.

Equilateral Triangle Construction Template

[color=#000000]In the applet below, use the tools of the limited toolbar to construct an [b]equilateral triangle that's inscribed in the circle shown.[/b] [br][br][i]When you're finished, show that the triangle you've constructed is indeed an equilateral triangle.[/i][/color]
– 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

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