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Formulas Involving Polygons - Lesson 7-3
Today, started by learning how polygons are classified by their number of sides...you already knew a lot of these - just make sure to memorize the ones you don't know!!
![Polygon Classification](polygonClassification.gif)
Next, we looked at the diagonals of polygons with different numbers of sides. We realized that by drawing as many diagonals as we could from one diagonal, that we could see a pattern...we can make n-2 triangles in a n-sided polygon. Given this information and the fact that the sum of the interior angles of a polygon is 180°, we came up with the following theorem:
![Polygon Sum Theorem](polygonSum.gif)
After that, we looked at exterior angles in a polygon. First, we looked at the exterior angles of a pentagon as shown below:
![Exterior Angles of a Pentagon](exteriorAngleSum1.gif)
Note that the sum of the exterior angles is 360°. Remember that I moved the sides of the pentagon around, thereby changing the measure of the exterior angles, but that the sum always stayed 360°. Next, we looked at a heptagon and found the same to be true. In fact, this is true of all polygons and can be written as a conjecture:
![Exterior Angle Sum](exteriorAngleSum2.gif)
Also remember the dilation I showed you...I shrunk down the heptagon, keeping the angles the same measure. This showed that the exterior angles formed a circle (therefore must sum to 360°. If you don't understand the following sequence of images, don't sweat it...just remember that the sum of the exterior angles is 360° no matter how many sides the polygon has:
![Exterior Angle Sum Dilation 1](exteriorAngleSum3.gif)
![Exterior Angle Sum Dilation 2](exteriorAngleSum4.gif)
![Exterior Angle Sum Dilation 3](exteriorAngleSum5.gif)
We then reviewed the following theorem...I showed some of you the derivation of this, but am not holding you responsible for it!
![Number of Diagonals](Theorem56.gif)
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