Geometric Notation: Points, Lines and Planes

The basic objects of geometry

Geometry describes positions and the objects joining them. A point represents an exact position. It has no size, even though it must be shown by a small dot on paper. Points are normally labelled with capital letters, such as $A$, $B$ and $C$.

A mathematical line is straight and continues for ever in both directions. A drawing can show only part of it, so arrowheads at both ends indicate that it continues. A line through points $A$ and $B$ may be written $\overleftrightarrow{AB}$.

A line segment is different: it is the finite part between two endpoints. The segment joining $A$ to $B$ is written $\overline{AB}$, although GCSE questions often use $AB$ when the context makes clear that they mean the segment or its length. Unlike a line, a segment can be measured. A ray starts at one endpoint and continues without end in one direction.

Planes and dimensions

A plane is a perfectly flat surface extending without end in all directions within that surface. A sheet of paper suggests part of a plane, but the mathematical plane has no boundary or thickness. Points and lines lying in one plane are coplanar.

This matters when describing parallel lines. In a flat diagram, every line is in the plane of the page. In three dimensions, however, two lines might pass at different levels and never meet without being parallel; such lines are not coplanar.

Vertices and edges

A vertex is a corner where sides or edges meet. Its plural is vertices, not “vertexes”. A square has four vertices. On a solid, several edges may meet at one vertex.

An edge is a finite line segment forming part of the boundary of a solid, commonly where two faces meet. A cube has edges, vertices and flat faces. An edge is therefore not an infinite line, even if it appears straight in a diagram. For a two-dimensional polygon, the boundary segments are usually called its sides.

Parallel lines

Two lines are parallel when they lie in the same plane, always remain the same perpendicular distance apart and never intersect. If the line through $A$ and $B$ is parallel to the line through $C$ and $$, write

$AB \parallel CD.$

Matching arrow marks drawn on the two lines show that they are parallel. A second pair of parallel lines may use matching double arrows so that the two pairs are not confused.

Be careful about two different uses of arrowheads. Arrowheads at the ends of one drawn line show that the line continues indefinitely. Matching arrow marks placed on two separate lines state that those lines are parallel.

Perpendicular lines

Two lines are perpendicular when they intersect to form an angle of exactly $90^\circ$. If $PQ$ is perpendicular to $RS$, write

$PQ \perp RS.$

A small square at the intersection marks the right angle. Perpendicular lines do not have to be horizontal and vertical: the whole diagram could be rotated and the angle between them would still be $90^\circ$.

Parallel and perpendicular describe relationships, not how lines happen to look. A diagram may not be drawn accurately, so mathematical markings and written statements provide the reliable information.