Name: ______Page 1

Straight and Symmetries Dialogue 6Reactions

Mathematician: Over the course of the year we have talked a lot about triangles and features that they have. What are some of the features we have focused on lately?
Student 1: Well we know that a triangle has an interior angle sum of 180 degrees.
Student 2: And that isosceles triangles have not only two congruent sides but also two congruent base angles.
Student 3: Oh! And we know that isosceles triangles have reflection symmetry and equilateral triangles have three reflection symmetries and three rotation symmetries.
Mathematician: Very Good. We have quite a list of properties that triangles possess. Today I’d like to use what we know about triangles to figure out what the interior angle sum of any polygon would be.
Student 2: Don’t we know that a quadrilateral has an interior angle sum of 360 degrees?
Mathematician: Ah yes I have heard that conjecture around the classroom. It happens to be true but we have not proved it yet.
Student 3: And that conjecture is only for quadrilaterals. I’m interested in seeing how to figure out the angle sum for any polygon.
Mathematician: All right, then lets get started. Consider the following polygon:

Student 1: Wait isn’t this a ______and don’t we already know that it has 360 degrees.
Mathematician: Yes, but we have not explained why this shape has an interior angle sum of 360 degrees. To do this we will utilize the idea of a diagonal. What happens to the shape if we draw a diagonal from one vertex to the opposite vertex?
Student 2: Then we can draw a line from A to C and from B to D. We would end up with this:


Student 3: Okay, I see how triangles will become important in our explanation because there are 4 triangles created by the two intersecting diagonals.
Student 1: Oh wait! I see 4 more triangles if we count ones that overlap. That gives us 8 potential triangles to work with. I just don’t see how this can help us explain why a ______has ______degrees.
Mathematician: These are very good points and you are right. If we consider both diagonals our situation becomes much more complicated. Notice before I actually said “What would happen if we draw a diagonal from one vertex to the opposite vertex?” As a mathematician I have to take each and every detail into consideration. I’ve trained myself to take the meaning from each and every word in a statement or definition. It often times makes the difference between a problem that is solvable and one that is unsolvable.
Student 1: So paying attention to detail, we only need to draw one diagonal and get:

Student 2: Of course, I see it now. By drawing one diagonal instead of two we now only have two triangles.
Student 1: And we already know that a triangle has 180 degrees, so two triangles is 360 degrees.
Student 3: And if the two triangles make up the ______then if I take all of the angles to the two triangles and add them they must be equal to the four angles of the ______.
Student 1: OK, so we now know that a quadrilateral has 360 degrees but I thought we would be able to tell what the interior angle sum of any polygon would be. If I draw a diagonal on a polygon that has more than 4 sides than I won’t end up with two triangles. See:




Student 1: From Hexagon ABCDEF I get a triangle, which I know has 180 degrees and a pentagon, which I do not know the angle measure for.
Mathematician: I see the dilemma… Well mathematicians have a habit of breaking down a problem into manageable pieces. This should help us here. When confronted with a new problem that I do not know how to do I try and break it into pieces that I do know how to solve. So for our pentagon dilemma (a shape that I do not know the interior angle measure). What if you drew another diagonal to break up the pentagon into more manageable pieces?
Student 1: Oh ok. Once I do that I will have a triangle and quadrilateral in the place of the pentagon. And if I do it again the quadrilateral will break into two more triangles and I would end up with:




Student 2: So the original hexagon ABCDEF is now broken into 4 triangles.
Student 3: This must mean that since a triangle has 180 degrees then a hexagon must have 4 times 180 degrees or 720 degrees.
Mathematician: You got it! Now lets see if we can generalize this even more by answering the following questions.

Answer the following questions based on the dialogue.

  1. Why would you not want the diagonals to cross when breaking a polygon into triangles? (Student 2 and top of page 2 for example)
  1. We know that a triangle has 180 degrees. Explain what is WRONG with the following explanation below:

This will be an explanation of why a triangle has 360 degrees. Consider triangle ABC below

If line AD is drawn then we have created two triangles. If each triangle has an angle sum of 180 degrees then the original must have an angle sum of twice 180 degrees. So a triangle has 360 degrees.

This explanation is wrong because…

  1. Can a triangle have any diagonals? Why or why not?
  1. Draw a pentagon and then break it up into triangles. How many did you get? What was thesum of the interior angle measure of the pentagon?
  1. Draw a heptagon and then break it up into triangles. How many did you get? What was thesum of the interior angle measure of the heptagon?
  1. Draw an octagon and then break it up into triangles. How many did you get? What was the sum of the interior angle measure of the octagon?
  1. Draw the craziest polygon you can think of (where it does not cross itself) and then break it up into triangles. How many sides is you polygon and how many triangles could you break it into?
  1. Do you notice any pattern between the number of sides a polygon has and the number of triangles that it can be cut into? What is the pattern?
  1. How could this pattern help you find the sum of the interior angles of a polygon?