Chapter 4: Polygon

4.1 Regular Polygons

A polygon is a closed 2D shape bounded by straight lines as its sides.

Regular vs Irregular Polygons

• Regular Polygon: A polygon where all sides are equal in length (congruent) and all interior angles are equal in measure. It possesses lines of symmetry equal to its number of sides $n$.
• Irregular Polygon: A polygon with sides of different lengths and/or interior angles of different measures.

Axis of Symmetry of Regular Polygons

For any regular polygon with $n$ sides, the number of axes of symmetry is equal to the number of sides $n$.

4.2 Interior Angles and Exterior Angles of Polygons

Sum of Interior Angles

By dividing an $n$-sided polygon into $(n - 2)$ non-overlapping triangles from one vertex, the total sum of interior angles can be calculated using the formula:

$$\text{Sum of interior angles} = (n - 2) \times 180^\circ$$

Interior Angle of a Regular Polygon

Since all interior angles of a regular $n$-sided polygon are equal, each interior angle is given by:

$$\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$$

Exterior Angles of a Polygon

An exterior angle is formed when a side of a polygon is extended outwards. An interior angle and its adjacent exterior angle lie on a straight line and add up to $180^\circ$:

$$\text{Interior Angle} + \text{Exterior Angle} = 180^\circ$$

Sum of Exterior Angles

For any polygon (regular or irregular), the sum of all its exterior angles is always $360^\circ$:

$$\text{Sum of exterior angles} = 360^\circ$$

Exterior Angle of a Regular Polygon

For a regular $n$-sided polygon, each exterior angle is equal and can be calculated by:

$$\text{Exterior Angle} = \frac{360^\circ}{n}$$

Determining the Number of Sides ($n$)

The number of sides $n$ of a regular polygon can be determined when the exterior angle or interior angle is known:

$$n = \frac{360^\circ}{\text{Exterior Angle}}$$

4.3 Problem Solving Involving Polygons

To solve complex geometric problems involving polygons:

1. Calculate missing interior or exterior angles using angle rules.
2. Use parallel line properties (alternate, corresponding, and interior angles on parallel lines) where applicable.
3. Apply triangle and quadrilateral angle properties to solve for unknown variables ($x, y, z$).