Embedded Files
Materials
Materials
Interior Angles in a Polygon worksheet, page 2 (one per person)
Straight edge (one per person)
Separating the Interior from the Exterior
Separating the Interior from the Exterior
Our final strategy utilizes the relationship between interior angles and exterior angles. Exterior angles are formed when a side is extended and an angle is formed from the extended side to the side adjacent to it. The exterior angles of a polygon add to 360 degrees. Understanding why this is the case can be explored by rotating through the exterior angles of a polygon in a manner similar to the 'Round About the Triangle activity.
Note: Advanced students may enjoy the challenge of proving that the sum of exterior angle measures of a polygon add to 360 degrees using induction.
The key to this strategy is that each exterior angle and corresponding interior angle form a supplementary angle pair as the extended side forms a straight angle. By finding the sum of all the interior/exterior angle pairs, we can solve for the sum of interior angle measures and using the fact that the sum of exterior angle measures is 360 degrees. The triangle is done as an example on the worksheet and students should be able to repeat the argument for the quadrilateral.
Note on Notation
Note on Notation
Carefully writing out the measures of each angle is quite tedious but we can simplify our notation simply by representing the sums of interior and exterior angle measures in a single variable; I do so below using words to make the same arguments with less writing.
Note: I've also been very careful to use consistent language and emphasize that we are finding the sum of angle measures and using the measure of angles notation in my expressions. An alternative notation to use is to reference the angles directly and identify the angles the sum is congruent to instead.
Pattern: Sum of Interior Angle Measures
Pattern: Sum of Interior Angle Measures
As we can see from our computations above, the sum of interior and exterior angle measures totals 180 degrees since each interior/exterior angle pair is supplementary. Since the sum of exterior angle measures is 360 degrees for every polygon, we can subtract this from the total so only the sum of interior angle measures remains. This leads us to a general expressions for the sum of interior angle measures of a polygon with n sides:
Final Note on Convex VS Concave Polygons
Final Note on Convex VS Concave Polygons
In our first two activities could be adjusted to account for concave polygons by decomposing them into concave polygons and using the arguments established to find the sum of interior angles and adding them together. However, in our third activity we need to be able to explain why the sum of exterior angle measures for concave polygons is also 360 degrees. Christopher Danielson has created an activity similar to that described in the 'Round About the Triangle activity for interior angles of a triangle to explain why the sum of exterior angles of a concave polygon is also 360 degrees. That activity is available online here.
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