Ray

Dependencies: Point–line axiom, Ruler axiom. A derived definition: choose an endpoint on a line and retain one of the two sides.

Definition

Let $O$ lie on line $\ell$. The Ruler axiom axiom orders the points of $\ell$ like the real numbers, so $O$ divides the line into two sides.

Choose one side. The point $O$ together with every point on that side forms a ray with endpoint $O$. If $A$ lies on the chosen side, the ray is denoted by $OA$.

A ray is half a line: closed at endpoint $O$ and extending without end in one direction. It has more extent than a segment and one fewer direction than a line.

Remarks

• Segment, ray, and line differ by extension. A segment has two endpoints and finite length; a ray has one endpoint and extends infinitely one way; a line extends infinitely both ways.
• Endpoint first, direction second. In $OA$, $O$ is the endpoint and $A$ only indicates direction. Rays $OA$ and $AO$ are different.
• Angles are built from rays. An angle consists of two rays with a common endpoint, and an angle bisector is itself a ray.