Quadrilateral

Dependency: Segment. A derived definition: four segments join end to end to enclose a four-sided figure.

Definition

Choose four points $A,B,C,D$ in order in the plane, with no three consecutive points collinear and no sides crossing.

The figure enclosed by segments $AB$, $BC$, $CD$, and $DA$ joined end to end is called quadrilateral $ABCD$. The points are its vertices, the segments are its sides, and $AC$ and $BD$ are its diagonals.

A quadrilateral has one more side than a triangle, and consequently is not rigid: four hinged bars can change shape. Additional constraints create the family of special quadrilaterals.

Remarks

• Opposite sides, opposite angles, and diagonals. Nonadjacent sides and nonadjacent angles are opposite pairs; diagonals join opposite vertices. These relationships organize most quadrilateral properties.
• Convex and concave. A quadrilateral is convex when both diagonals lie inside it, and concave when part of a diagonal lies outside. Unless stated otherwise, the special quadrilaterals here are convex.
• Adding conditions creates a family. Two pairs of parallel sides give a parallelogram; exactly one pair gives a trapezoid; four right angles give a rectangle; and four equal sides give a rhombus.
• Angle sum $360^\circ$. One diagonal divides a quadrilateral into two triangles, so its interior angles total $2\times180^\circ=360^\circ$ (Polygon angle sum = (n − 2)·180°).