Polygons & Angles

Tim Brzezinski, Donovan, Feb 25, 2018

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

Table of Contents

  1. Classifying Polygons
    1. Convex vs. Concave
    2. Equilateral Action!!!
    3. Equiangular Action!!!
    4. Regular Polygon Action!
  2. Polygon Interior & Exterior Angle Sum Theorems
    1. Copy of Triangle Angle Theorems (V1)
    2. Copy of Quadrilateral Angle Theorems
    3. Copy of Pentagon Angle Theorems
    4. Copy of Hexagon Angle Theorems
    5. Copy of Heptagon Angle Theorems
    6. Copy of Octagon Angle Theorems
  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. Convex vs. Concave
  • 2. Equilateral Action!!!
  • 3. Equiangular Action!!!
  • 4. Regular Polygon Action!

1.1.

Convex vs. Concave

In the app below, move the vertices of the shown polygon around. Be sure to explore the triangle, quadrilateral, and pentagon. Then, answer the questions that follow.
According to the app, how would you describe what it means for a polygon to be [color=#ff7700][b]convex[/b][/color]? How would you describe what it means for a polygon to be [color=#1e84cc][b]concave[/b][/color]?
Is it ever possible for a triangle to be [color=#1e84cc][b]concave[/b][/color]? Why or why not?
– 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. Copy of Triangle Angle Theorems (V1)
  • 2. Copy of Quadrilateral Angle Theorems
  • 3. Copy of Pentagon Angle Theorems
  • 4. Copy of Hexagon Angle Theorems
  • 5. Copy of Heptagon Angle Theorems
  • 6. Copy of Octagon Angle Theorems

2.1.

Copy of Triangle Angle Theorems (V1)

[color=#000000]Interact with the applet below for a few minutes.  [br]Then, answer the questions that follow.  [br][br][/color][i][color=#980000]Be sure to change the locations of the triangle's WHITE VERTICES each time before you drag the slider!!! [/color][/i]
[b][color=#980000]Questions:[/color][/b][br][color=#000000]  [br]1) From your observations, what is the sum of the measures of the interior angles of[br]    [i]any triangle?  [br][/i][br]2) When working with the triangle's exterior angles, did any of these transformations change the measures of the [/color][color=#6aa84f]green[/color][color=#000000] or [/color][color=#999999]gray[/color][color=#000000] angles? [br][br]3) From your observations, what is the sum of the measures of the exterior angles of[br]    [i]any triangle?  [/i][/color]
– Donovan, 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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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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