Section 2.4

Angles in Polygons

Finding the Sum of the Interior Angles of any Polygon

1. Draw any polygon

2. Draw all of the polygon's diagonals form one vertex, thus dividing the polygon into triangles

3. Count the number of triangles

4. Since we know that the sum of the angles of any triangle = 180⁰ , if we multiply the number of triangles formed by 180⁰ we will find the sum of the interior angles of any polygon.

By doing this I divide a square unto two triangle.

We know that the sum of the interior angles of a triangle is 180 degrees.

Therefore 2 triangles x 180⁰ = 360⁰

Therefore the sum of the interior angles of a square is equal to 360⁰

What would be the sum of the interior angles of the following polygons?

1. pentagon

2. hexagon

3. heptagon

1. octagon

2. n-gon

In a polygon with n sides, the sum of the interior angle measures is

(n-2) 180⁰

Finding the sum of the Exterior Angles of a Polygon

If one side of a polygon is extended, the extended ray and the adjacent side of the polygon form an exterior angle of that polygon.

At each vertex of a polygon an interior angle and its adjacent exterior angle form a linear pair.

In our example the interior angle BCD and its adjacent exterior angle BCE form a linear pair.

Therefore the sum of the measure of /_ BCD and the measure of /_ BCE equal 180 degrees.

Activity:

1. Take a ruler and draw a triangle, a quadrilateral, and a pentagon.

2. Using the ruler extend the sides of each polygon.

3. Use a protractor to measure all the exterior angles of each polygon.

4. Add together the measure of all the exterior angles of each polygon to find the sum of the measure of the exterior angles of a triangle, a quadrilateral, and a pentagon.

5. What did you find that the sum of the measure of the exterior angles of a triangle, a quadrilateral, and a pentagon have in common?

The sum of the measures of the exterior angles of any polygon is 360 degrees.

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