Quadrilaterals

A quadrilateral is any closed figure bounded by four line segments, giving four sides, four vertices and four angles. Joining two opposite vertices gives a diagonal, which splits the quadrilateral into two triangles. Because the angles of a triangle add to 180 degrees and there are two triangles, the four angles of any quadrilateral always add up to 360 degrees. This single fact, the angle sum property, is the first tool you reach for whenever angles are involved.

Quadrilaterals come in a family. A trapezium has one pair of parallel sides. A parallelogram has both pairs of opposite sides parallel. A rectangle is a parallelogram whose angles are all right angles. A rhombus is a parallelogram with all four sides equal. A square is the special member that is both a rectangle and a rhombus. Every square is a rectangle and a rhombus, but not the other way round, so the shapes nest inside one another like boxes within boxes.

Drawing a diagonal of a parallelogram creates two congruent triangles, and that congruence proves everything else. In a parallelogram opposite sides are equal, opposite angles are equal, and the diagonals bisect each other (each cuts the other into two equal halves). These are the properties you assume once you know a figure is a parallelogram, and they are stated as theorems so they can be relied on in proofs.

Each property also works backwards, as a converse, giving you a test. A quadrilateral is a parallelogram if its opposite sides are equal, or if its opposite angles are equal, or if its diagonals bisect each other. There is also a powerful shortcut: if just one pair of opposite sides is both equal and parallel, the figure must be a parallelogram. These converses let you start from a few facts and conclude the shape, which is how most exam proofs are built.

The chapter ends with a beautiful result about triangles. If you join the mid-points of two sides of a triangle, the segment you draw is parallel to the third side and exactly half as long. Its converse is equally useful: a line through the mid-point of one side, drawn parallel to a second side, must bisect the third side. The mid-point theorem connects triangles back to quadrilaterals, because joining the mid-points of any quadrilateral always produces a parallelogram.