Rays

A ray is a part of a line that starts at a specific point and extends infinitely in one direction.

It originates from a point \( O \), known as the origin or endpoint, which divides the line into two parts, with only one part being considered.

The points on the ray, apart from the origin, are called its interior points.

A ray can be labeled with a lowercase letter, similar to lines (e.g., r, s, etc.), or using the notation \( \overrightarrow{OA} \), where \( O \) represents the origin and \( A \) is any point that determines the direction in which the ray extends.

The concept of a ray invites us to think about infinity. While a ray might seem "incomplete" because it lacks one endpoint, its endless extension in a single direction distinguishes it from a line segment (which is finite) and a line (which is infinite in both directions).

To visualize a ray, imagine a beam of light coming from a flashlight: it has a starting point (the flashlight) and continues infinitely in one direction. This is a helpful way to conceptualize a ray.

Any line can be split into two rays by choosing a specific point on the line as the origin \( O \).

The two resulting rays extend in opposite directions from \( O \) and are referred to as opposite rays.

In simpler terms, two opposite rays together form the line that includes the origin \( O \) and all points belonging to both rays \( r \) and \( s \).

For example, consider a line with a point \( O \) on it.

The two opposite rays \( r \) and \( s \) can be represented as:

• \( \overrightarrow{OA} \), extending in one direction;
• \( \overrightarrow{OB} \), extending in the opposite direction.

Although these two rays share the same origin, they have no points in common other than \( O \). However, when taken together, they form the entire line \( r \).

Rays are fundamental in geometry, especially for defining angles. For instance, an angle is formed by two rays \( r \) and \( s \) that share a common origin \( O \).