Polygons & Angles ( s n attar)

Tim Brzezinski, Shamashuddin Attar, Sep 30, 2017

Contains interactive applets pertaining to interior and exterior angles of polygons.

Table of Contents

  1. Classifying Polygons
    1. Equiangular Action!!!
    2. Equilateral Action!!!
    3. Regular Polygon Action!
    4. Convex vs. Concave
  2. Polygon Interior & Exterior Angle Sum Theorems
    1. Triangle Angle Theorems
    2. Triangle Angle Theorems (V2)
    3. Quadrilateral Angle Theorems
    4. Pentagon Angle Theorems
    5. Hexagon Angle Theorems
    6. Heptagon Angle Theorems
    7. Octagon Angle Theorems
    8. Exploring Polygon Angles: Triangle through Octagon
  3. Polygon Exterior Angle Sum Theorems (Easier to See)
    1. Polygons: Exterior Angles
    2. Polygons: Exterior Angles - REVAMPED
    3. Triangle: Exterior Angles
    4. Exterior Angles of a Triangle
    5. Quadrilateral: Exterior Angles
    6. Exterior Angles of a Quadrilateral
    7. Pentagon: Exterior Angles
    8. Exterior Angles of a Pentagon
    9. Exterior Angles of a Hexagon
    10. Exterior Angles of a Heptagon
    11. Exterior Angles of an Octagon
  4. Other & Older Versions of Applets in Chapter 1
    1. Triangle Angle Theorems (V3)
    2. Triangle Interior & Exterior Angle Sum Theorems (II)
    3. Quadrilateral Interior & Exterior Angle Sum Theorems (V2)
    4. Quadrilateral Interior & Exterior Angle Sum Theorems (V2)
    5. Pentagon Interior & Exterior Angle Sum Theorems
    6. Hexagon Interior & Exterior Angle Sum Theorems
    7. Heptagon Interior & Exterior Angle Sum Theorems
    8. Octagon Interior & Exterior Angle Sum Theorems
    9. Exterior Angles of Polygons--Proofs without Words
  5. Basic Illustrations
    1. Triangle Angle Sum Theorem
  6. Higher-Level Problems with Polygons and Angles
    1. Cut-The-Knot-Action (5)!
    2. Rhombus Action + Sequel = GoGeometry Action 7!

1.

Classifying Polygons

  • 1. Equiangular Action!!!
  • 2. Equilateral Action!!!
  • 3. Regular Polygon Action!
  • 4. Convex vs. Concave

1.1.

Equiangular Action!!!

The applet below contains an [b]equiangular pentagon[/b]. [br]Interact with the applet for a minute, then answer the question that follows. [br][br]Be sure to change the locations of the WHITE POINTS and/or the[color=#1e84cc] [b]BLUE POINT[/b][/color] during your investigation.
EQUIANGULAR PENTAGON
1.
What does it mean for a polygon to be an [b][color=#ff00ff]equiangular polygon[/color][/b]?
– Tim Brzezinski

1.2.

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1.3.

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1.4.

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2.

Polygon Interior & Exterior Angle Sum Theorems

  • 1. Triangle Angle Theorems
  • 2. Triangle Angle Theorems (V2)
  • 3. Quadrilateral Angle Theorems
  • 4. Pentagon Angle Theorems
  • 5. Hexagon Angle Theorems
  • 6. Heptagon Angle Theorems
  • 7. Octagon Angle Theorems
  • 8. Exploring Polygon Angles: Triangle through Octagon

2.1.

Triangle Angle Theorems

Interact with the app below for a few minutes.  [br]Then, answer the questions that follow.  [br][br]Be sure to change the locations of this triangle's vertices each time [i]before[/i] you drag the slider!
What is the [b]sum of the measures of the interior angles of this triangle? [/b]
What is the [b]sum of the measures of the exterior angles [/b]of this triangle?
– Tim Brzezinski

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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3.

Polygon Exterior Angle Sum Theorems (Easier to See)

  • 1. Polygons: Exterior Angles
  • 2. Polygons: Exterior Angles - REVAMPED
  • 3. Triangle: Exterior Angles
  • 4. Exterior Angles of a Triangle
  • 5. Quadrilateral: Exterior Angles
  • 6. Exterior Angles of a Quadrilateral
  • 7. Pentagon: Exterior Angles
  • 8. Exterior Angles of a Pentagon
  • 9. Exterior Angles of a Hexagon
  • 10. Exterior Angles of a Heptagon
  • 11. Exterior Angles of an Octagon

3.1.

Polygons: Exterior Angles

The exterior angles of a triangle, quadrilateral, and pentagon are shown, respectively, in the applets below. [br][br]You can control the size of a colored exterior angle by using the slider with matching color. [br]Feel free to move the vertices of these polygons anywhere you'd like. [br][br][b]Note:[/b] [br]For the [b]quadrilateral[/b] & [b]pentagon[/b], the last two applets work best if these polygons are kept [b]convex.[/b][br]If you don't remember what this term means, [url=https://www.geogebra.org/m/knnPDMR3]click here for a refresher[/url].
Exterior Angles of a Triangle
Exterior Angles of a Quadrilateral
Exterior Angles of a Pentagon
What do you notice? What is common about the measures of the exterior angles of any one of these polygons?
Do you think what you've observed for the triangle, quadrilateral, and pentagon above will also hold true for a hexagon, heptagon, and octagon? [br][br]Create a new GeoGebra file and do some investigating to informally test your hypotheses!
– Tim Brzezinski

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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3.8.

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3.9.

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3.10.

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3.11.

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4.

Other & Older Versions of Applets in Chapter 1

  • 1. Triangle Angle Theorems (V3)
  • 2. Triangle Interior & Exterior Angle Sum Theorems (II)
  • 3. Quadrilateral Interior & Exterior Angle Sum Theorems (V2)
  • 4. Quadrilateral Interior & Exterior Angle Sum Theorems (V2)
  • 5. Pentagon Interior & Exterior Angle Sum Theorems
  • 6. Hexagon Interior & Exterior Angle Sum Theorems
  • 7. Heptagon Interior & Exterior Angle Sum Theorems
  • 8. Octagon Interior & Exterior Angle Sum Theorems
  • 9. Exterior Angles of Polygons--Proofs without Words

4.1.

Triangle Angle Theorems (V3)

[b]Directions:[/b][br][br]Interact with the applet below for a few minutes.[br]Each time you move the slider, do so slowly. Pay careful attention to what you observe each time.[br][b][color=#1551b5]Be sure to move the [/color][color=#444]black vertices[/color] [color=#1551b5]of the triangle around each time after you reset the slider! [/color] [/b][br][br][i]Be sure to answer the questions that will appear in the applet. [/i]
– 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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4.9.

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5.

Basic Illustrations

  • 1. Triangle Angle Sum Theorem

5.1.

Triangle Angle Sum Theorem

– Tim Brzezinski

6.

Higher-Level Problems with Polygons and Angles

  • 1. Cut-The-Knot-Action (5)!
  • 2. Rhombus Action + Sequel = GoGeometry Action 7!

6.1.

Cut-The-Knot-Action (5)!

Creation of this applet was inspired by a [url=https://twitter.com/CutTheKnotMath/status/839945580088602629]tweet[/url] from [url=https://twitter.com/CutTheKnotMath]Alexander Bogomolny[/url]. [br][br]In the applet below, a [b][color=#6aa84f]regular pentagon[/color][/b] and a [color=#bf9000][b]regular decagon[/b][/color] share a common side. [br][br][color=#ff00ff][b]What is the measure of the pink angle? [/b][/color] [br][br][color=#0000ff][b]How can you formally prove what this applet informally illustrates? [/b][/color]
Quick (Silent) Demo
– Tim Brzezinski

6.2.

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