0

Polygons & Angles

Tim Brzezinski, Sep 30, 2015

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

Classifying Polygons
1. Convex vs. Concave
2. Equilateral Action!!!
3. Equiangular Action!!!
4. Regular Polygon Action!
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
9. Lambourghini Turning Angles
Polygon Exterior Angle Sum Theorems (Easier to See)
1. Polygons: Exterior Angles
2. Polygons: Exterior Angles - REVAMPED
3. Exterior Angles (Revisited)
4. Exterior Angles (Revisited) V2
5. Triangle: Exterior Angles
6. Exterior Angles of a Triangle
7. Quadrilateral: Exterior Angles
8. Exterior Angles of a Quadrilateral
9. Pentagon: Exterior Angles
10. Exterior Angles of a Pentagon
11. Exterior Angles of a Hexagon
12. Exterior Angles of a Heptagon
13. Exterior Angles of an Octagon
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
Basic Illustrations
1. Triangle Angle Sum Theorem
Higher-Level Problems with Polygons and Angles
1. Cut-The-Knot-Action (5)!
2. Rhombus Action + Sequel = GoGeometry Action 7!

Information

Classifying Polygons

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

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 convex? How would you describe what it means for a polygon to be concave?

Is it ever possible for a triangle to be concave? Why or why not?

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
9. Lambourghini Turning Angles

Triangle Angle Theorems

Interact with the app below for a few minutes. Then, answer the questions that follow. Be sure to change the locations of this triangle's vertices each time before you drag the slider!

What is the sum of the measures of the interior angles of this triangle?

What is the sum of the measures of the exterior angles of this triangle?

Polygon Exterior Angle Sum Theorems (Easier to See)

Polygons: Exterior Angles

The exterior angles of a triangle, quadrilateral, and pentagon are shown, respectively, in the applets below. You can control the size of a colored exterior angle by using the slider with matching color. Feel free to move the vertices of these polygons anywhere you'd like. Note: For the quadrilateral & pentagon, the last two applets work best if these polygons are kept convex. If you don't remember what this term means, click here for a refresher.

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? Create a new GeoGebra file and do some investigating to informally test your hypotheses!

3.2.

3.3.

3.4.

3.5.

3.6.

3.7.

3.8.

3.9.

3.10.

3.11.

3.12.

3.13.

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)

Directions: Interact with the applet below for a few minutes. Each time you move the slider, do so slowly. Pay careful attention to what you observe each time. Be sure to move the black vertices of the triangle around each time after you reset the slider! Be sure to answer the questions that will appear in the applet.

4.2.

4.3.

4.4.

4.5.

4.6.

4.7.

4.8.

4.9.

5.

Basic Illustrations

1. Triangle Angle Sum Theorem

5.1.

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!

6.1.

Cut-The-Knot-Action (5)!

Creation of this applet was inspired by a tweet from Alexander Bogomolny. In the applet below, a regular pentagon and a regular decagon share a common side. What is the measure of the pink angle? How can you formally prove what this applet informally illustrates?

0