Key Points

A polygon is a simple closed curve made up of only line segments. Polygons can be classified as convex, where all diagonals lie inside, or concave, where at least one diagonal lies outside.

A regular polygon is both equiangular (all angles are equal in measure) and equilateral (all sides are of equal length). If a polygon is not regular, it is called an irregular polygon.

The sum of the measures of the four interior angles of any quadrilateral is always $360^\circ$.

The sum of the measures of the exterior angles of any convex polygon, with one angle taken at each vertex, is always $360^\circ$. This is true regardless of the number of sides.

For a regular polygon with $n$ sides, the measure of each exterior angle is $\frac{360^\circ}{n}$. Conversely, the number of sides can be found by $n = \frac{360^\circ}{\text{exterior angle}}$.

A parallelogram is a quadrilateral with two pairs of parallel sides. Its key properties are: (1) Opposite sides are equal. (2) Opposite angles are equal. (3) Diagonals bisect each other.

The adjacent angles in a parallelogram are supplementary, which means their sum is $180^\circ$. If $\angle A$ and $\angle B$ are adjacent, then $m\angle A + m\angle B = 180^\circ$.

A trapezium is a quadrilateral that has exactly one pair of parallel sides. If the non-parallel sides are equal in length, it is called an isosceles trapezium.

A kite is a quadrilateral with two distinct pairs of equal-length consecutive sides. Its diagonals are perpendicular to each other, and one of the diagonals bisects the other.

A rhombus is a parallelogram with all four sides of equal length. A key property is that its diagonals are perpendicular bisectors of one another.

A rectangle is a parallelogram where every angle is a right angle ($90^\circ$). A key property is that its diagonals are of equal length and they bisect each other.

A square is a rectangle with equal sides, making it a regular quadrilateral. Its diagonals are equal, bisect each other, and are perpendicular to each other.

A square is a special kind of rectangle and rhombus. A rectangle and a rhombus are special kinds of parallelograms. A parallelogram is a special kind of trapezium.