Geometry: Exterior and Interior Angles of Polygons

Study Notes

Measures of Exterior and Adjacent Interior Angles of a Convex Polygon

Exterior Angle Theorem

The exterior angle of a polygon is formed by one side of the polygon and the extension of an adjacent side.
The sum of exterior angles of any convex polygon is always 360 degrees.
Each exterior angle can be calculated as the difference between 180 degrees and the corresponding interior angle.

Interior Angle Sum

The sum of the interior angles ( S ) of a convex polygon with ( n ) sides is given by the formula: [ S = (n - 2) \times 180^\circ ]
For example, a triangle (3 sides) has an interior angle sum of ( 180^\circ ), while a quadrilateral (4 sides) has ( 360^\circ ).

Convex Polygon Properties

A convex polygon has all interior angles less than 180 degrees.
The sides of a convex polygon do not intersect except at vertices.
For a regular convex polygon (all sides and angles equal):
- Each interior angle can be calculated using: [ \text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n} ]
- Each exterior angle: [ \text{Exterior Angle} = \frac{360^\circ}{n} ]

Angle Relationships in Polygons

Each pair of adjacent interior and exterior angles is supplementary (adds up to 180 degrees).
The interior angle can be found using: [ \text{Interior Angle} = 180^\circ - \text{Exterior Angle} ]
The relationship between the number of sides ( n ), the interior angle ( I ), and the exterior angle ( E ): [ I + E = 180^\circ \quad \text{and} \quad I = 180^\circ - E ]

Calculating Angles

To calculate the exterior angle of a regular polygon:
1. Determine ( n ) (number of sides).
2. Use the formula: [ E = \frac{360^\circ}{n} ]
To find the interior angle:
1. Use ( n ) in the formula: [ I = \frac{(n - 2) \times 180^\circ}{n} ]
For irregular polygons, use known angles to apply the interior angle sum formula and calculate unknown angles.

Exterior Angle Theorem

An exterior angle is created by one side of a polygon and the extension of an adjacent side.
The total of all exterior angles in any convex polygon equals 360 degrees.
Each exterior angle can be determined by subtracting the interior angle from 180 degrees.

Interior Angle Sum

The formula for the sum of interior angles ( S ) in a convex polygon with ( n ) sides is ( S = (n - 2) \times 180^\circ ).
A triangle has a sum of interior angles equal to 180 degrees; a quadrilateral has a sum of 360 degrees.

Convex Polygon Properties

Convex polygons feature all interior angles measuring less than 180 degrees.
Sides of a convex polygon intersect only at their vertices.
For regular convex polygons, each interior angle is calculated as ( \text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n} ).
Each exterior angle in a regular polygon is calculated by ( \text{Exterior Angle} = \frac{360^\circ}{n} ).

Angle Relationships in Polygons

Adjacent interior and exterior angles are supplementary, meaning they add up to 180 degrees.
The interior angle can be found using ( \text{Interior Angle} = 180^\circ - \text{Exterior Angle} ).
The relationships among the number of sides ( n ), the interior angle ( I ), and the exterior angle ( E ) can be expressed as ( I + E = 180^\circ ) and ( I = 180^\circ - E ).

Calculating Angles

To calculate the exterior angle of a regular polygon:
- Identify the number of sides ( n ).
- Use the formula ( E = \frac{360^\circ}{n} ).
To derive the interior angle of a regular polygon:
- Plug in ( n ) into ( I = \frac{(n - 2) \times 180^\circ}{n} ).
For irregular polygons, apply the interior angle sum formula with known angles to solve for unknown angles.

Description

Explore the properties of exterior and interior angles in convex polygons. Understand the Exterior Angle Theorem and the calculations for interior angle sums. This quiz will challenge your knowledge of polygon geometry and angle measures.