Geometry and Measurement - Properties of Quadrilaterals and Polygons

Polygon Interior Angles: The sum of the interior angles of a polygon depends on the number of sides $n$. For any polygon, you can divide the shape into $n-2$ triangles by drawing diagonals from a single vertex. Since each triangle accounts for $180^{\circ}$, the total sum is $(n-2) \times 180^{\circ}$. For example, a pentagon ($n=5$) can be split into 3 triangles, totaling $540^{\circ}$.

Regular Polygons: A regular polygon is both equilateral (all sides are the same length) and equiangular (all interior angles are equal). Visually, these shapes have a center point from which all vertices are equidistant. Because all angles are equal, the measure of a single interior angle in a regular polygon is the total sum divided by $n$.

Exterior Angles: An exterior angle is formed by extending one side of a polygon. The interior angle and its adjacent exterior angle always lie on a straight line, meaning they sum to $180^{\circ}$. For any convex polygon, no matter the number of sides, the sum of all exterior angles is always $360^{\circ}$. This can be visualized by imagining a person walking around the perimeter and turning at each corner; by the time they return to the start, they have made one full $360^{\circ}$ rotation.

Properties of Parallelograms: A parallelogram is a quadrilateral with two pairs of parallel sides. Key visual and geometric properties include: opposite sides are equal in length, opposite angles are equal, and consecutive angles (angles next to each other) are supplementary, adding up to $180^{\circ}$. Furthermore, the diagonals of a parallelogram bisect each other, meaning they cross at their exact midpoints.

Special Quadrilaterals (Rhombus and Square): A rhombus is a parallelogram with four equal sides. Its diagonals are perpendicular ($90^{\circ}$), forming four congruent right-angled triangles inside the shape. A square is a regular quadrilateral, possessing the properties of both a rectangle (four $90^{\circ}$ angles) and a rhombus (four equal sides and perpendicular diagonals).

Trapeziums and Kites: A trapezium (or trapezoid) has at least one pair of parallel sides. An isosceles trapezium has non-parallel sides of equal length and equal base angles, making it visually symmetrical. A kite has two pairs of adjacent equal sides. Visually, one diagonal of a kite acts as a line of symmetry, and the diagonals always intersect at a right angle ($90^{\circ}$).