Envision_Geom_notes_6-1

Section 6-1: The Polygon Angle-Sum Theorems

Learning Target

A) I can find the sums of the measures of the exterior angles and interior angles of polygons.

Sums of Angle Measures in Convex Polygons

A convex polygon has all sides pointing outwards, (0<x<180’)
For polygons with a few sides, polygons can be broken down into triangles by connecting one vertex to all other vertices using diagonals.
Each triangle has an angle sum of 180°.
The formula to find the total number of degrees in any convex polygon is:
- (n – 2) ⋅ 180
- Here, n represents the number of sides.
The number of triangles formed is exactly 2 less than the number of sides of the polygon.
For finding a degree in a quadrilateral use ((n-2)x180))/n to find an interior angle
to find an exterior angle, divide 360 by the number of sides.
For a pentagon, the calculation becomes ((5-2)x180))/5 for the interior angle, which simplifies to 108 degrees, while the exterior angle would be 360/5, resulting in 72 degrees.
For a hexagon, the interior angle can be calculated using ((6-2)x180))/6, yielding 120 degrees, and the exterior angle would be 360/6, which equals 60 degrees.
If it is an irregular polygon, add up all the sides and then subract that from (n-2)180

Example: Convex Decagon

Decagon = 10 sides
- Calculation: (10 – 2) ⋅ 180 = 8 ⋅ 180 = 1440°

Regular Polygon Interior Angle Measures

To find the measure of one interior angle in a regular polygon (where all sides and angles are equal), use the following formula:
- Measure of one interior angle = Total degrees in the polygon ÷ Number of angles in the polygon
- (n-2)x180=k, k/n= 1 interior angle
Example: Regular Decagon
- Total degrees in decagon = 1440°
- Calculation: 1440 ÷ 10 = 144°

Polygon Exterior Angle-Sum Theorem

The sum of the measures of all exterior angles of a convex polygon (one exterior angle at each vertex) is always 360°. This holds true regardless of the number of sides the polygon has. ( 360/number of sides = the measure of each exterior angle, where n is the number of sides of the polygon.)

Key Points:

Definition of Exterior Angle: An exterior angle is formed by one side of a polygon and the extension of an adjacent side.
Exterior Angle Calculation: For any polygon, you can calculate an individual exterior angle by subtracting the measure of its corresponding interior angle from 180°.
- For example, if the interior angle of a regular polygon is known, the exterior angle can be found using the formula:
[ \text{Exterior Angle} = 180° - \text{Interior Angle} ]
Regular Polygons: In regular polygons (where all sides and angles are equal), each exterior angle can be calculated by dividing the total sum of the exterior angles (360°) by the number of sides (n):
[ \text{Measure of One Exterior Angle} = \frac{360°}{n} ]
Examples:
- For a regular pentagon (5 sides), each exterior angle would be:
[ \frac{360°}{5} = 72° ]
- For a regular octagon (8 sides), each exterior angle would be:
[ \frac{360°}{8} = 45° ]
Geometric Proof: The relationship of the sum of exterior angles being 360° can be visualized by rotating around the polygon. By traveling around the polygon, every turn made at each vertex effectively completes a full circular rotation of 360°.

Understanding the exterior angles of polygons is essential in fields such as architecture and engineering, where accurate measurements and calculations are critical to design and construction. The exterior angle properties also play a crucial role in polygons being used in various applications, reinforcing the fundamental nature of geometry in real-world scenarios.