⬡ Polygons & Quadrilaterals

• Interior angle sum: for an $n$-sided polygon the angles add to $(n-2)\times 180^\circ$ — so triangle $180^\circ$, quadrilateral $360^\circ$, pentagon $540^\circ$, hexagon $720^\circ$ (each side adds $180^\circ$).

• Exterior angle sum: always $360^\circ$ for any convex polygon, whatever $n$ is.

• Regular polygon (all angles equal): each exterior angle $=\dfrac{360^\circ}{n}$ and each interior angle $=\dfrac{(n-2)\times 180^\circ}{n}$. To find the sides quickly, use $n=\dfrac{360^\circ}{\text{exterior angle}}$.

• Diagonals: a polygon has $\dfrac{n(n-3)}{2}$ diagonals.

At every vertex, interior $+$ exterior $=180^\circ$ — the link that turns one into the other.

The four interior angles always add to $360^\circ$. The special shapes add their own rules:

• Parallelogram (covers rhombus, rectangle, square): opposite angles equal, adjacent angles supplementary, diagonals bisect each other.

• Rectangle: all angles $90^\circ$ and diagonals equal. Rhombus: all sides equal, diagonals perpendicular and bisect the angles. Square: both at once.

• Trapezium: one pair of parallel sides, so a diagonal gives equal alternate (“Z”) angles.

• Cyclic quadrilateral (vertices on a circle): each pair of opposite angles sums to $180^\circ$, and the exterior angle equals the interior opposite angle.