There are three activities here, all to do with geometry and proof. If you are reading this page on the web, all the images can be enlarged by clicking on them.
We start with the most important tools of a mathematician:
The paper here is a sheet of A4 paper. What shape is it? (What do we mean by "shape"?) You may find it helpful to know that when an A4 sheet is cut into two identical rectangles (each being A5) each of these is the same shape as the original A4 sheet.
We are now going to fold the paper. I suggest you get a sheet of paper of your own and try it for yourself. Think carefully as you follow the process...
First fold a corner to opposite corner:
We have folded a pentagon. It seems symmetrical. You might like to think about why it is symmetrical.
In all questions like this you should be thinking about a "perfect" A4-sheet of paper, not a imperfect sheet cut by a slightly inaccurate machine and folded by a very inaccurate human being. We are, after all, mathematicians, not engineers or physicists!
The pentagon just folded, whilst symmetrical, is certainly not regular. (What does it mean for a pentagon to be regular?) Here are some more folds, starting with folding the figure just made in half, and then unfolding, to make a centre line.
Is this pentagon regular?
You may have seen a demonstration of this fact involving cutting out a triangle and pasting its angles onto a straight line.
Is this a proof?
If you are unsure about whether this demonstration is entirely valid or subject to experimental error, or whether it adequately proves the fact for all "perfect" triangles then you should provide your own water-tight mathematical proof.
If points A B and C lie on a circle with centre O, where the line segment AC is a diameter of the circle (i.e., passes through the circle's centre), what can you say about the angle ABC? (Hint: draw the line segment OB)
Now suppose A, B, C still lie on the circle, but this time none of the sides of triangle ABC is a diameter. What can you say about angles AOB and ACB? (Hint: draw the line segment OC.)
Richard Kaye