Thursday 1 December class (updated)

Topics:

• Checking that the angles add up correctly in the "algebraic" proof of the Pythagorean Theorem from Math is Fun.

Clearly the sides of the interior square match up with the hypotenuse of the right triangle - that is how we constructed the square. The only question is whether the angles add up correctly where the corners of that square touch the sides of the big square - we are asking if the sides of the big square are really straight lines, as they appear to be.

We can check this by using two facts: 

1) In any right triangle, the other two angles add up to 90 degrees (they are complimentary). 

2) The three angles that meet at the corner of the interior square consist of a right angle (from the square), and the two other non-right angles from the right triangle. They are labeled A and B in the picture below (look at the top edge of the big square).

So the total of those three angles is 90 degrees plus angle A plus angle B, which is 9- degrees plus 90 degrees = 180 degrees, a straight line, as it appears to be. So all is well, the big square really is a square and we can compute its area by squaring the length of its side.

• Polygons, regular polygons, and introduction to regular tilings (tessellations) of the plane.

Polygons (that link also gives the names of many of the polygons)
Regular polygons and their exterior and interior angles.
Tiling the plane with regular polygons (Regular tilings; we will discuss semi-regular tilings next time)

Homework:
Find the measures of the exterior and interior angles for the following regular polygons:
1) the regular nonagon (9 sides)
2) the regular dodecagon (12 sides)

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