Understanding Quadrilaterals - Polygons and their Classification

Polygon Definition: A polygon is a simple closed plane figure bounded by three or more line segments. Visually, it is a shape where the lines never cross and the starting point meets the ending point to enclose a space. Common examples include triangles, quadrilaterals, and pentagons.

Classification by Sides: Polygons are named based on the number of sides ($n$) they possess. For example, a polygon with $n=3$ is a triangle, $n=4$ is a quadrilateral, $n=5$ is a pentagon, $n=6$ is a hexagon, and $n=10$ is a decagon.

Convex and Concave Polygons: A polygon is convex if all its interior angles are less than $180^\circ$ and all its diagonals lie entirely inside the figure. A concave polygon (or re-entrant polygon) has at least one interior angle greater than $180^\circ$, appearing as if one or more vertices are 'pushed inward'. Visually, you can draw a line between two points inside a concave polygon that passes outside the shape.

Regular and Irregular Polygons: A regular polygon is both equilateral (all sides are equal) and equiangular (all angles are equal). For example, an equilateral triangle and a square are regular polygons. If any side or angle differs from the others, the polygon is irregular. Visually, regular polygons look perfectly symmetrical.

Diagonals: A diagonal is a line segment connecting two non-consecutive vertices of a polygon. In a triangle, no diagonals can be drawn ($0$ diagonals), while in a quadrilateral, two diagonals can be drawn, often forming an 'X' shape within the four-sided figure.

Interior Angle Sum Property: The sum of the interior angles of a polygon with $n$ sides is always constant for that specific $n$. This property is derived from the fact that any $n$-sided polygon can be divided into $(n-2)$ triangles. Since each triangle has a sum of $180^\circ$, the total sum is $(n-2) \times 180^\circ$.

Exterior Angle Property: An exterior angle is formed by extending one side of a polygon. The sum of the measures of the exterior angles of any convex polygon, taken in order, is always $360^\circ$, regardless of the number of sides. Visually, if you rotate through each exterior angle, you would complete exactly one full turn.