Learning Math: Geometry
Polygons Part C: Definitions and Proof (45 minutes)
Session 3, Part C
In this part:
- Definitions
- Understanding Definitions
- Dividing Polygons into Triangles
- Triangles in Convex Polygons
Definitions
Definitions and proof both play essential roles in mathematics. Definition is crucial, because you need to know what something is (and what it’s not) before you can make conjectures about it and prove these conjectures.
When mathematicians create a definition, they strive to be concise — to communicate a lot of information in a few words. This can make reading and understanding mathematical definitions difficult. This is further complicated by the fact that more than one definition may work for a given object. You may find yourself coming across an unfamiliar definition for a familiar object, and somehow you have to make sense of it.
A benefit of mathematical definitions is that you’ll never find the circularity associated with dictionary definitions. For example:
Webster’s attempt to define “dimension”
Dimension: | Any measurable extent, such as length, width, and depth. |
Extent: | The space, amount, or degree to which a thing extends; size; length; breadth. |
Measurement: | Extent, quantity, or size as determined by measuring. |
Size: | That quality of a thing that determines how much space it occupies; dimensions; extent. |
Length: | The measure of how long a thing is; the greatest of the two or three dimensions of anything; extent in space. |
Here’s a mathematical definition of a geometric property called convexity. The word may be familiar to you, but try to focus just on the definition provided for this activity:
A figure is convex if, for every pair of points within the figure, the segment connecting the two points lies entirely within the figure.
Problem C1:
Use the definition above to make sense of the notion of “convex figures.” What do they look like? Can you describe what they look like in your own words? Take whatever steps are necessary for you to understand the mathematical definition. Describe the steps you took to understand the definition. How did you make sense of it for yourself?
Definitions and Proof adapted with the permission of Educational Development Center, Inc. This material was created under NSF Grant Number ES1-9818736. Opinions expressed are those of the author and not necessarily shared by the funder. For further information, visit http://www2.edc.org/makingmath/.
Definitions of dimension, extent, measurement, size, and length taken From Webster’s New World College Dictionary, Fourth Edition. Copyright © 2000, 1999 by Wiley Publishing, Inc. All rights reserved. Reproduced here by permission of Wiley Publishing, Inc.
Understanding Definitions
People go about understanding mathematical definitions in different ways; the steps they take may vary. Here’s one way:
Step 1. | Read the definition more than once. |
Step 2. | Identify what “things” the definition is talking about. |
Step 3. | Generate a test case. |
Step 4. | Determine if the example fits the definition. |
Step 5. | Find examples that do not fit the definition. |
Step 6. | Try to generalize the examples to create an image of the full concept. (In the case of convex figures, you might think about curved figures, three-dimensional figures, etc.) |
Convex, like many other mathematical ideas, has several different, but equivalent definitions. Not every proposed definition will work, though, and some definitions are better than others — they may be more clear, use fewer words, or be easier to test.
Problem C2
Which of these definitions work for convex polygons? A polygon is convex if and only if…
- all diagonals lie in the interior of the polygon.
- the perimeter is larger than the length of the longest diagonal.
- every diagonal is longer than every side.
- the perimeter of the polygon is the shortest path that encloses the entire shape.
- the largest interior angle is adjacent to the longest side.
- none of the lines that contain the sides of the polygon pass through its interior.
- every interior angle is less than 180°.
- the polygon is not concave.
Dividing Polygons into Triangles
You may know that the sum of the angles in a triangle is 180°. Can you prove it? In Session 4, we’ll look at one mathematical argument for which the result is true. For now, we’ll assume that it’s true (based on some strong evidence) and look at some consequences of that fact. This is how mathematicians often work: They assume an intermediate result, called a lemma. If the lemma turns out to be useful in proving other results, they go back and try to prove that the lemma itself is true.
Problem C3
Polygons with any number of sides can be divided up into triangles. Here are a few examples:
Draw several other examples of polygons divided into triangles for polygons of varying numbers of sides. Be sure not to use just regular polygons, and be sure not to use just convex polygons.
Problem C4
How would you divide the polygons below into triangles?
Make sure your method works for regular and irregular polygons, and also for convex and concave polygons.
Take It Further
Problem C5
Describe a method so that, given any polygon, you are able to divide it into triangles.
Triangles in Convex Polygons
There are a few different methods that will work for dividing a polygon into triangles. One method is particularly convenient because, for polygons with the same number of sides, you get the same number of triangles. Here’s an outline of the method for convex polygons (minor changes are necessary if you work with a concave polygon):
- Pick any vertex to start.
- Connect that vertex to every other vertex, except the two that are adjacent to it.
Problem C6
Use the method above or your own method, and fill in the table below. Remember that we are assuming that there are 180° in a triangle. When you click “Show Answers,” the filled-in table will appear below the problem. Scroll down the page to see it.
Number of Sides of the Polygon |
Number of Triangles Formed |
Sum of the Angles in the Polygon |
3 | 1 | 180° |
4 | ||
5 | ||
6 | ||
7 | ||
n |
Problem C7
Write a convincing mathematical argument to explain why your result for the sum of the angles in an n-gon is correct. Note 5
Notes
Note 5
For problem C7, take the time to summarize the argument as a group or reflect deeply if working alone. Don’t forget that we are assuming the angle sum of a triangle, but have an argument (not really a formal proof) that you can make (n – 2) triangles.
Solutions
Problem C1
Answers will vary. One possible way of describing convex figures is that they are figures that are not dented, or that a rubber band stretched around the figure will touch the figure entirely.
Problem C2
Definitions (a), (d), (f), (g), and (h) are all equivalent descriptions of convex polygons. Statement (b) is true for all polygons, so it would apply to concave ones as well (triangle inequality!). Statement (c) is not true for all convex polygons. For example, try drawing a convex quadrilateral with one very long side; this side will probably be longer than the shortest diagonal of the quadrilateral. Statement (e) certainly does not describe convex polygons. (Consider an obtuse triangle which is convex but whose longest side is opposite to its largest angle.)
Problem C3
Answers will vary. You can find some more examples in Problem C4.
Problem C4
The answers may vary for the two figures on the right.
Problem C5
If an n-sided polygon is convex, we can pick a vertex and connect it to all other vertices, thereby creating n – 2 triangles. If the polygon is concave, we pick a vertex corresponding to an interior angle greater than 180° and connect that vertex to all the vertices so that all the resulting diagonals are in the interior of the polygon. Repeat the process if necessary. This has the effect of subdividing the original polygon into some number of convex polygons. We then divide each of the convex polygons into triangles. In the end, we will end up with n – 2 triangles.
Problem C6
Number of Sides of the Polygon | Number of Triangles Formed | Sum of the Angles in the Polygon |
---|---|---|
3 | 1 | 180˚ |
4 | 2 | 360˚ |
5 | 3 | 540˚ |
6 | 4 | 720˚ |
7 | 5 | 900˚ |
n | n-2 | (n-2) • 180˚ |
Problem C7
Assuming that the interior of an n-sided polygon can be divided into n – 2 triangles, observe that all of the angles of the triangles actually make up the interior angles of the polygon. This is true since the vertices of all the triangles coincide with the vertices of the polygon. Therefore, since each triangle contributes 180° to the overall sum of the angles, the sum is (n – 2) • 180°.