Understanding Quadrilaterals - Angle Sum Property of a Polygon

A polygon is a simple closed plane figure bounded by three or more line segments. For example, a quadrilateral is a four-sided polygon, which can be visualized as a closed shape with four straight edges and four corners (vertices).

The Interior Angle Sum Property states that for any convex polygon with $n$ sides, the sum of the measures of all its interior angles is $(n - 2) \times 180^{\circ}$. This can be visualized by picking one vertex and drawing all possible diagonals from it to divide the polygon into $(n-2)$ triangles, each having an angle sum of $180^{\circ}$.

The Exterior Angle Sum Property states that the sum of the measures of the exterior angles (taken in order) of any convex polygon is always $360^{\circ}$, regardless of the number of sides. If you imagine walking around the perimeter of the polygon and turning at each corner, you would make one full $360^{\circ}$ rotation to return to your starting direction.

A Regular Polygon is a polygon that is both equilateral (all sides are equal) and equiangular (all interior angles are equal). Examples include an equilateral triangle or a square. Because all angles are equal, each interior angle measures $\frac{(n - 2) \times 180^{\circ}}{n}$ and each exterior angle measures $\frac{360^{\circ}}{n}$.

At every vertex of a polygon, the interior angle and the exterior angle are supplementary, meaning their sum is $180^{\circ}$. Visually, if you extend one side of a polygon to form a straight line, the interior angle and the newly formed exterior angle together form a linear pair.

A Convex Polygon is a polygon where every interior angle is less than $180^{\circ}$. In these shapes, all diagonals lie entirely within the interior of the polygon. In contrast, a Concave Polygon has at least one interior angle greater than $180^{\circ}$ (a reflex angle), causing the shape to look 'caved in' and resulting in at least one diagonal lying outside the polygon.