Congruence (CO) Cluster

JohnCMorris, Nov 14, 2017

MAFS.912.G-CO.1.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MAFS.912.G-CO.1.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). MAFS.912.G-CO.1.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. MAFS.912.G-CO.1.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. MAFS.912.G-CO.1.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. MAFS.912.G-CO.2.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. MAFS.912.G-CO.2.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. MAFS.912.G-CO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. MAFS.912.G-CO.3.9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segments endpoints. MAFS.912.G-CO.3.10 Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180 triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MAFS.912.G-CO.3.11 Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. MAFS.912.G-CO.4.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. MAFS.912.G-CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Table of Contents

  1. MAFS.912.G-CO.1.1
    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. MAFS.912.G-CO.1.2, 1.3, 1.4, 1.5
    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. MAFS.912.G-CO.2.6
    1. Are the triangles congruent (part 2)?
    2. Are the triangles congruent (part 3)?
  4. MAFS.912.G-CO.2.7
    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. MAFS.912.G-CO.2.8
    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. MAFS.912.G-CO.3.9
    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. Interesting Coplanar Locations (1)
    35. A Special Theorem: Part 1 (V1)
    36. A Special Theorem: Part 2 (V1)
    37. A Special Theorem: Part 2 (V2)
    38. A Special Theorem: Part 2 (V3)
    39. Animation 57
    40. A Special Theorem: Part 2 (V4)
    41. PBT (Part 1) - Quick
    42. PBT (Part 2) - Quick
    43. Where in This City? (1)
    44. CT Home Depots
    45. Where in NYC? (VA)
    46. Where in NYC? (VB)
    47. Where in NYC? (VC)
    48. Angle Bisector Theorem (Part 1)
    49. Another Special Theorem: Part 1 (V1)
    50. Another Special Theorem: Part 2 (V1)
    51. Another Special Theorem: Part 1 (V2)
    52. Animation 58
    53. Another Special Theorem: Part 1 (V3)
    54. Effortless Trisections
    55. Animation 134
    56. Parallel Lines Congruent Segments Theorem
    57. Parallel Lines Proportionality Theorem
    58. Animation 135
  7. MAFS.912.G-CO.3.10
    1. Triangle Area Action! (V1)
    2. Triangle Area Action!!! (V2)
    3. Animation 175
    4. Area of a Triangle (Discovery)
    5. 4 Triangles Action!!!
    6. Animation 179
    7. Triangle Angle Theorems
    8. Animation 171
    9. Triangle Angle Theorems (V2)
    10. Animation 2
    11. Triangle Angle Theorems (V3)
    12. Transformation Proof 3
    13. Triangle Angle Sum Theorem
    14. Obvious Corollary
    15. Animation 140
    16. Triangle Exterior Angle
    17. Animation 102
    18. Triangle: Exterior Angles
    19. Quadrilateral: Exterior Angles
    20. Pentagon: Exterior Angles
    21. Polygons: Exterior Angles
    22. Polygons: Exterior Angles - REVAMPED
    23. Exploring Polygon Angles: Triangle through Octagon
    24. Angles of a Right Triangle
    25. Animation 104
    26. Acute Angles of a Right Triangle
    27. Animation 129
    28. Animation 22
    29. Isosceles Triangle Theorem: Discovery Lab
    30. A Special Triangle & Its Properties (I)
    31. Animation 8
    32. Isosceles Triangle Investigation
    33. Converse of IST (V1)
    34. Animation 82
    35. Animation 23
    36. Another Special Triangle and its Properties (II)
    37. Animation 27
    38. Converse of IST (II)
    39. Animation 84
    40. Isosceles Right Triangle: Quick Investigation
    41. Special Right Triangle (I)
    42. Animation 59
    43. Special Right Triangle (II)
    44. Another Special Right Triangle (Guided Discovery)
    45. Animation 42
    46. Special Line through Triangle V1 (Theorem Discovery)
    47. Special Line through Triangle V2 (Theorem Discovery)
    48. Triangle Midsegment Action!
    49. Animation 141
    50. Midsegment of a Triangle
    51. Triangle Inequality: Discovery Lesson
    52. Triangle Side Possibilities? (V2)
    53. Triangle Inequality
    54. Triangle Inequality (II)
    55. Inequalities in 1 Triangle
    56. Points of Concurrency: Investigation
    57. Orthocenter Exploration
    58. Circumcenter (& Questions)
    59. Circumcenter & Circumcircle Action!
    60. Animation 28
    61. Midpoint of HYP
    62. Animation 46
    63. Incenter Exploration (A)
    64. Incenter Exploration (B)
    65. Incenter & Incircle Action!
    66. Animation 17
    67. Triangle Medians: Quick Investigation
    68. Medians Centroid Theorem (Proof without Words)
    69. Medians and Centroid Dance
    70. Animation 25
  8. MAFS.912.G-CO.3.11
    1. Parallelogram: Area
    2. Animation 202
    3. Parallelogram Construction Template & Discovery Lab
    4. Parallelogram Template (Scaffolded Discovery)
    5. Parallelogram Template
    6. Parallelogram: Theorem 1
    7. Animation 123
    8. Parallelogram: Theorem 2
    9. Animation 124
    10. Parallelogram: Theorem (3)
    11. Animation 125
    12. Parallelograms (I)
    13. Animation 75
    14. Animation 80
    15. Animation 6
    16. What Kind of Quadrilateral is This? (Part 1)
    17. What Kind of Quadrilateral is This? (Part 2)
    18. What Kind of Quadrilateral is This? (Part 3)
    19. What Kind of Quadrilateral is This? (Part 4)
    20. Special Quadrilateral: Theorem 1
    21. Animation 184
    22. Special Quadrilateral: Theorem 2
    23. Animation 185
    24. Special Quadrilateral: Theorem 3
    25. Animation 186
    26. Special Quadrilateral: Theorem 4
    27. Animation 187
    28. Congruent Opposite Sides of a Quadrilateral
    29. Quadrilateral With Opposite Angles Congruent
    30. What Kind of Quad?
    31. Quadrilateral Investigation (I)
    32. Rhombus Action!
    33. Animation 11
    34. Rhombus Template (Scaffolded Discovery)
    35. Rhombus Template with Investigation Questions
    36. Rectangle Action!
    37. Animation 31
    38. Rectangle Template (Scaffolded Investigation)
    39. Rectangle Template with Investigation Questions
    40. Square Action!
    41. Animation 41
    42. Square Template (Scaffolded Investigation)
    43. Square Template with Investigation Questions
    44. Trapezoids: Median Investiagtion
    45. Trapezoid Median (Midsegment) Action!
    46. Animation 67
    47. Trapezoid Median: 2 Discoveries
    48. Isosceles Trapezoid Action!
    49. Animation 64
    50. Animation 35
    51. Isosceles Trapezoid Template (Scaffolded Discovery)
    52. Isosceles Trapezoid Template
    53. Animation 49
    54. Kite Action!
    55. Animation 26
    56. Kite Template: Scaffolded Investigation
    57. Kite Template with Investigation Questions
    58. Quad Midpoints Action!
    59. Animation 52
  9. MAFS.912.G-CO.4.12
    1. Congruent Segments Construction (I)
    2. Isosceles Triangle (Quick Construction Technique) V1
    3. Animation 199
    4. Equilateral Triangle Construction (Dynamic Illustration)
    5. Animation 201
    6. Congruent Angle: Construction
    7. Congruent Angle Construction Action!
    8. Animation 196
    9. Angle Bisector Construction Action!
    10. Animation 197
    11. Alternate Angle Bisector Construction
    12. Animation 117
    13. Angle Bisector: Construction
    14. Angle Bisector (Basic Exercise)
    15. Perpendicular Line: Construction 1
    16. Perpendicular Line: Construction 2
    17. Perpendicular Bisector: Construction
    18. Parallel Line: Constructions
    19. Congruent Triangle: Construction
    20. Rectangle Construction Exercise
    21. Rectangle Construction Exercise: Revamped (V1)
    22. Rectangle Construction Exercise: Revamped (V2)
    23. Rhombus Construction Exercise: V1
    24. Rhombus Construction Exercise: V2
    25. Other Constructions: 1
    26. Other Constructions: 2
    27. Centroid Construction Challenge #1
    28. Possible Solution for Centroid Challenge #1
    29. Centroid Construction Challenge #2
    30. Possible Solution for Centroid Construction Challenge #2
    31. Centroid Construction Challenge #3
    32. Possible solution to Centroid Construction Challenge #3
    33. Triangle Construction Challenge #1
    34. Possible solution to Triangle Construction Challenge #1
    35. Triangle Consruction Challenge #2
    36. Possible solution to Triangle Construction Challenge #2
    37. Triangle Construction Challenge #3
    38. Potential solution for Triangle Construction Challenge #3
    39. Geometric Mean Illustration
    40. Animation 211
    41. Geometric Mean: Construction (VA)
    42. Geometric Mean: Construction (VB)
  10. MAFS.912.G-CO.4.12
    1. Equilateral Triangle Construction Template
    2. Square Construction Template
    3. Regular Hexagon Construction Template

1.

MAFS.912.G-CO.1.1

  • 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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2.

MAFS.912.G-CO.1.2, 1.3, 1.4, 1.5

  • 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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3.

MAFS.912.G-CO.2.6

  • 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.

MAFS.912.G-CO.2.7

  • 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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MAFS.912.G-CO.2.8

  • 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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MAFS.912.G-CO.3.9

  • 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. Interesting Coplanar Locations (1)
  • 35. A Special Theorem: Part 1 (V1)
  • 36. A Special Theorem: Part 2 (V1)
  • 37. A Special Theorem: Part 2 (V2)
  • 38. A Special Theorem: Part 2 (V3)
  • 39. Animation 57
  • 40. A Special Theorem: Part 2 (V4)
  • 41. PBT (Part 1) - Quick
  • 42. PBT (Part 2) - Quick
  • 43. Where in This City? (1)
  • 44. CT Home Depots
  • 45. Where in NYC? (VA)
  • 46. Where in NYC? (VB)
  • 47. Where in NYC? (VC)
  • 48. Angle Bisector Theorem (Part 1)
  • 49. Another Special Theorem: Part 1 (V1)
  • 50. Another Special Theorem: Part 2 (V1)
  • 51. Another Special Theorem: Part 1 (V2)
  • 52. Animation 58
  • 53. Another Special Theorem: Part 1 (V3)
  • 54. Effortless Trisections
  • 55. Animation 134
  • 56. Parallel Lines Congruent Segments Theorem
  • 57. Parallel Lines Proportionality Theorem
  • 58. Animation 135

6.1.

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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6.31.

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6.32.

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6.33.

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6.34.

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6.35.

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6.36.

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6.37.

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6.38.

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6.39.

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6.40.

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6.41.

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6.42.

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6.43.

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6.44.

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6.45.

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6.46.

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6.47.

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6.48.

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6.49.

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6.50.

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6.51.

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6.52.

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6.53.

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6.54.

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6.55.

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6.56.

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6.57.

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6.58.

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7.

MAFS.912.G-CO.3.10

  • 1. Triangle Area Action! (V1)
  • 2. Triangle Area Action!!! (V2)
  • 3. Animation 175
  • 4. Area of a Triangle (Discovery)
  • 5. 4 Triangles Action!!!
  • 6. Animation 179
  • 7. Triangle Angle Theorems
  • 8. Animation 171
  • 9. Triangle Angle Theorems (V2)
  • 10. Animation 2
  • 11. Triangle Angle Theorems (V3)
  • 12. Transformation Proof 3
  • 13. Triangle Angle Sum Theorem
  • 14. Obvious Corollary
  • 15. Animation 140
  • 16. Triangle Exterior Angle
  • 17. Animation 102
  • 18. Triangle: Exterior Angles
  • 19. Quadrilateral: Exterior Angles
  • 20. Pentagon: Exterior Angles
  • 21. Polygons: Exterior Angles
  • 22. Polygons: Exterior Angles - REVAMPED
  • 23. Exploring Polygon Angles: Triangle through Octagon
  • 24. Angles of a Right Triangle
  • 25. Animation 104
  • 26. Acute Angles of a Right Triangle
  • 27. Animation 129
  • 28. Animation 22
  • 29. Isosceles Triangle Theorem: Discovery Lab
  • 30. A Special Triangle & Its Properties (I)
  • 31. Animation 8
  • 32. Isosceles Triangle Investigation
  • 33. Converse of IST (V1)
  • 34. Animation 82
  • 35. Animation 23
  • 36. Another Special Triangle and its Properties (II)
  • 37. Animation 27
  • 38. Converse of IST (II)
  • 39. Animation 84
  • 40. Isosceles Right Triangle: Quick Investigation
  • 41. Special Right Triangle (I)
  • 42. Animation 59
  • 43. Special Right Triangle (II)
  • 44. Another Special Right Triangle (Guided Discovery)
  • 45. Animation 42
  • 46. Special Line through Triangle V1 (Theorem Discovery)
  • 47. Special Line through Triangle V2 (Theorem Discovery)
  • 48. Triangle Midsegment Action!
  • 49. Animation 141
  • 50. Midsegment of a Triangle
  • 51. Triangle Inequality: Discovery Lesson
  • 52. Triangle Side Possibilities? (V2)
  • 53. Triangle Inequality
  • 54. Triangle Inequality (II)
  • 55. Inequalities in 1 Triangle
  • 56. Points of Concurrency: Investigation
  • 57. Orthocenter Exploration
  • 58. Circumcenter (& Questions)
  • 59. Circumcenter & Circumcircle Action!
  • 60. Animation 28
  • 61. Midpoint of HYP
  • 62. Animation 46
  • 63. Incenter Exploration (A)
  • 64. Incenter Exploration (B)
  • 65. Incenter & Incircle Action!
  • 66. Animation 17
  • 67. Triangle Medians: Quick Investigation
  • 68. Medians Centroid Theorem (Proof without Words)
  • 69. Medians and Centroid Dance
  • 70. Animation 25

7.1.

Triangle Area Action! (V1)

In the applet below, [color=#38761d][b]a triangle is shown[/b][/color]. [br][br]You can change the sizes of the [b][color=#1e84cc]blue angle[/color][/b] and [color=#ff00ff][b]pink angle[/b][/color] by using the sliders in the lower-left corner. You can also move the white points anywhere you'd like at any time. [br][br]Interact with the applet below for a few minutes.[br]Then answer the questions that follow.
Area of a Triangle = ?
1.
In this applet's dynamics, the triangle was transformed into another figure. What kind of figure is it? How does the applet imply your classification is correct?
2.
How does the [b]area of the newer figure[/b] compare with the [b]area of the original triangle[/b]?
3.
How does the [b]BASE [/b]of the newer figure compare with the [b]BASE[/b] of the original triangle?
4.
How does the [color=#9900ff][b]HEIGHT[/b][/color] of the newer figure compare with the [color=#444444][b]HEIGHT[/b][/color] of the original triangle?
5.
Given your responses to (1) - (4) above, describe how you can find the area of [b][color=#38761d]ANY TRIANGLE. [/color][/b]
Quick (Silent) Demo
– Tim Brzezinski

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7.6.

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7.7.

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7.8.

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7.9.

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7.10.

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7.11.

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7.12.

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7.13.

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7.14.

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7.15.

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7.16.

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7.17.

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7.18.

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7.19.

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7.20.

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7.21.

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7.22.

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7.23.

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7.24.

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7.25.

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7.26.

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7.27.

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7.28.

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7.29.

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7.30.

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7.31.

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7.32.

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7.33.

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7.34.

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7.35.

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7.36.

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7.37.

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7.38.

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7.39.

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7.40.

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7.41.

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7.42.

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7.43.

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7.44.

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7.45.

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7.46.

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7.47.

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7.48.

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7.49.

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7.50.

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7.51.

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7.52.

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7.53.

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7.54.

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7.55.

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7.56.

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7.57.

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7.58.

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7.59.

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7.60.

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7.61.

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7.62.

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7.63.

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7.64.

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7.65.

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7.66.

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7.67.

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7.68.

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7.69.

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7.70.

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8.

MAFS.912.G-CO.3.11

  • 1. Parallelogram: Area
  • 2. Animation 202
  • 3. Parallelogram Construction Template & Discovery Lab
  • 4. Parallelogram Template (Scaffolded Discovery)
  • 5. Parallelogram Template
  • 6. Parallelogram: Theorem 1
  • 7. Animation 123
  • 8. Parallelogram: Theorem 2
  • 9. Animation 124
  • 10. Parallelogram: Theorem (3)
  • 11. Animation 125
  • 12. Parallelograms (I)
  • 13. Animation 75
  • 14. Animation 80
  • 15. Animation 6
  • 16. What Kind of Quadrilateral is This? (Part 1)
  • 17. What Kind of Quadrilateral is This? (Part 2)
  • 18. What Kind of Quadrilateral is This? (Part 3)
  • 19. What Kind of Quadrilateral is This? (Part 4)
  • 20. Special Quadrilateral: Theorem 1
  • 21. Animation 184
  • 22. Special Quadrilateral: Theorem 2
  • 23. Animation 185
  • 24. Special Quadrilateral: Theorem 3
  • 25. Animation 186
  • 26. Special Quadrilateral: Theorem 4
  • 27. Animation 187
  • 28. Congruent Opposite Sides of a Quadrilateral
  • 29. Quadrilateral With Opposite Angles Congruent
  • 30. What Kind of Quad?
  • 31. Quadrilateral Investigation (I)
  • 32. Rhombus Action!
  • 33. Animation 11
  • 34. Rhombus Template (Scaffolded Discovery)
  • 35. Rhombus Template with Investigation Questions
  • 36. Rectangle Action!
  • 37. Animation 31
  • 38. Rectangle Template (Scaffolded Investigation)
  • 39. Rectangle Template with Investigation Questions
  • 40. Square Action!
  • 41. Animation 41
  • 42. Square Template (Scaffolded Investigation)
  • 43. Square Template with Investigation Questions
  • 44. Trapezoids: Median Investiagtion
  • 45. Trapezoid Median (Midsegment) Action!
  • 46. Animation 67
  • 47. Trapezoid Median: 2 Discoveries
  • 48. Isosceles Trapezoid Action!
  • 49. Animation 64
  • 50. Animation 35
  • 51. Isosceles Trapezoid Template (Scaffolded Discovery)
  • 52. Isosceles Trapezoid Template
  • 53. Animation 49
  • 54. Kite Action!
  • 55. Animation 26
  • 56. Kite Template: Scaffolded Investigation
  • 57. Kite Template with Investigation Questions
  • 58. Quad Midpoints Action!
  • 59. Animation 52

8.1.

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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8.11.

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8.14.

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8.15.

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8.16.

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8.17.

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8.18.

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8.19.

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8.20.

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8.21.

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8.23.

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8.24.

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8.25.

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8.26.

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8.27.

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8.28.

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8.29.

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8.30.

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8.31.

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8.32.

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8.33.

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8.34.

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8.35.

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8.36.

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8.37.

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8.38.

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8.39.

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8.41.

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8.42.

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8.43.

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8.44.

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8.48.

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8.50.

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8.51.

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8.52.

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8.53.

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8.54.

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8.55.

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8.56.

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8.57.

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8.59.

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9.

MAFS.912.G-CO.4.12

  • 1. Congruent Segments Construction (I)
  • 2. Isosceles Triangle (Quick Construction Technique) V1
  • 3. Animation 199
  • 4. Equilateral Triangle Construction (Dynamic Illustration)
  • 5. Animation 201
  • 6. Congruent Angle: Construction
  • 7. Congruent Angle Construction Action!
  • 8. Animation 196
  • 9. Angle Bisector Construction Action!
  • 10. Animation 197
  • 11. Alternate Angle Bisector Construction
  • 12. Animation 117
  • 13. Angle Bisector: Construction
  • 14. Angle Bisector (Basic Exercise)
  • 15. Perpendicular Line: Construction 1
  • 16. Perpendicular Line: Construction 2
  • 17. Perpendicular Bisector: Construction
  • 18. Parallel Line: Constructions
  • 19. Congruent Triangle: Construction
  • 20. Rectangle Construction Exercise
  • 21. Rectangle Construction Exercise: Revamped (V1)
  • 22. Rectangle Construction Exercise: Revamped (V2)
  • 23. Rhombus Construction Exercise: V1
  • 24. Rhombus Construction Exercise: V2
  • 25. Other Constructions: 1
  • 26. Other Constructions: 2
  • 27. Centroid Construction Challenge #1
  • 28. Possible Solution for Centroid Challenge #1
  • 29. Centroid Construction Challenge #2
  • 30. Possible Solution for Centroid Construction Challenge #2
  • 31. Centroid Construction Challenge #3
  • 32. Possible solution to Centroid Construction Challenge #3
  • 33. Triangle Construction Challenge #1
  • 34. Possible solution to Triangle Construction Challenge #1
  • 35. Triangle Consruction Challenge #2
  • 36. Possible solution to Triangle Construction Challenge #2
  • 37. Triangle Construction Challenge #3
  • 38. Potential solution for Triangle Construction Challenge #3
  • 39. Geometric Mean Illustration
  • 40. Animation 211
  • 41. Geometric Mean: Construction (VA)
  • 42. Geometric Mean: Construction (VB)

9.1.

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

9.2.

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9.3.

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9.4.

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9.12.

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9.14.

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9.17.

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9.18.

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9.20.

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9.21.

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9.23.

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9.24.

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9.25.

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9.26.

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9.27.

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9.28.

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9.29.

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9.30.

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9.31.

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9.32.

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9.33.

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9.34.

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9.35.

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9.36.

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9.38.

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9.39.

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9.40.

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9.41.

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9.42.

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10.

MAFS.912.G-CO.4.12

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

10.1.

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

10.2.

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10.3.

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Information