Polygons

From one vertex, draw diagonals to split an $n$-sided polygon into $n-2$ triangles. Each triangle has angles summing to $180^{\circ }$.

So the interior angles of an $n$-gon sum to $(n-2)\times 180^{\circ }$: a pentagon $540^{\circ }$, a hexagon $720^{\circ }$, a decagon $1440^{\circ }$.

In a $\text{regular}$ polygon, all sides and angles are equal, so each interior angle is the sum divided by $n$.

An exterior angle is formed by extending one side; it and the interior angle at that vertex lie on a straight line, so they sum to $180^{\circ }$.

The exterior angles of any polygon sum to $360^{\circ }$.

For a regular polygon, each exterior angle is $\frac{360^{\circ }}{n}$, so $n=\frac{360^{\circ }}{\text{exterior angle}}$.

$\text{Square}$: four equal sides, four right angles, diagonals equal and bisect at right angles.

$\text{Rectangle}$: opposite sides equal, four right angles, diagonals equal and bisect each other.

$\text{Parallelogram}$: opposite sides parallel and equal, opposite angles equal, diagonals bisect each other.

$\text{Rhombus}$: four equal sides, opposite angles equal, diagonals bisect at right angles. $\text{Kite}$: two pairs of adjacent equal sides, one pair of equal opposite angles, diagonals perpendicular. $\text{Trapezium}$: one pair of parallel sides.

Combine polygon facts with other angle facts: tessellations, shapes meeting at a point, or polygons joined along an edge.

Shapes tessellate around a point only if their angles there sum to $360^{\circ }$.