What Is a Ray?
A ray is a part of a line that starts at a single point and extends forever in one direction. You can think of it as a “half-line”: it has one endpoint and then continues without stopping.
Ray notation: \(\overrightarrow{AB}\)
The symbol \(\overrightarrow{AB}\) is read as “ray AB.” It means:
- Endpoint: the ray starts at point A.
- Direction: it passes through point B and continues beyond B forever.
The order of letters matters: \(\overrightarrow{AB}\) and \(\overrightarrow{BA}\) are usually different rays because they start at different endpoints and go in different directions.
How to recognize a ray in a drawing
- Look for a point where the figure “starts” (the endpoint).
- Look for an arrow showing it continues in one direction.
- If points are marked on it, the ray goes from the endpoint through those points and beyond.
In sketches, a ray is often drawn as a point with a line extending from it and an arrow at the far end.
Rays and Angle Formation
Angles are formed by two rays that share the same endpoint. That shared endpoint is the vertex of the angle.
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For example, if two rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) share endpoint A, they form an angle with vertex A. The “opening” of the angle depends on the directions from A through B and from A through C.
| Object | What to look for |
|---|---|
| Ray | One endpoint, one direction |
| Angle | Two rays with a common endpoint (vertex) |
Collinear Points and Rays on the Same Line
Points are collinear if they lie on the same straight line. Collinearity is important for rays because it helps you decide whether two rays are the same, different, or opposite.
Using a diagram to test collinearity
In a diagram, points are collinear if you could place a ruler so that it passes through all the points in a single straight path. If one point is “off” that straight path, the points are not collinear.
Collinearity and “lies on” language
When a point lies on a ray, it must be:
- On the same line as the ray, and
- On the correct side of the endpoint (in the ray’s direction).
Example idea: If B is on \(\overrightarrow{AC}\), then A, C, and B are collinear, and B is located in the direction from A through C (not behind A).
Same Ray vs. Different Rays
Two rays can share an endpoint and still be different. To decide whether \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are the same ray or different rays, check these conditions:
When are they the same ray?
\(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are the same ray if:
- They have the same endpoint A,
- Points B and C are collinear with A, and
- B and C are on the same side of A (both in the same direction from A).
In that case, B and C both lie on the same “half” of the line starting at A.
When are they different rays?
They are different rays if:
- B and C are not on the same line with A (not collinear), or
- They point in different directions from A.
Opposite Rays
Opposite rays are two rays that:
- Share the same endpoint,
- Are collinear (lie on the same line), and
- Extend in exactly opposite directions.
If \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are opposite rays, then A is between B and C on a straight line. Together, opposite rays form a straight line (a straight angle of 180°).
How to recognize opposite rays in drawings
- Find two rays with the same endpoint.
- Check whether they lie on one straight line.
- See whether one goes left (or one direction) while the other goes right (the opposite direction) from the endpoint.
A common visual clue: it looks like a straight line with a point marked somewhere in the middle as the shared endpoint, with arrows extending in both directions.
Mini-Lab: Classifying Rays from Plotted Points
In this mini-lab, you will create your own diagram and decide whether two rays are the same, different, or opposite.
Materials
- Paper
- Pencil
- Ruler (recommended)
Step-by-step
Draw point A. Place it near the center of your page.
Plot point B. Put B somewhere to the right of A. Draw the ray
\(\overrightarrow{AB}\): draw a line starting at A, passing through B, and add an arrow beyond B.Plot point C in three different trials (do these one at a time):
- Trial 1 (same ray): Place C on the same line as A and B, beyond B (so the order is A–B–C). Draw
\(\overrightarrow{AC}\). Decide: are\(\overrightarrow{AB}\)and\(\overrightarrow{AC}\)the same ray? - Trial 2 (opposite rays): Place C on the same line as A and B, but on the other side of A (so the order is C–A–B). Draw
\(\overrightarrow{AC}\). Decide: are the rays opposite? - Trial 3 (different rays): Place C so that A, B, and C are not collinear (for example, above the line through A and B). Draw
\(\overrightarrow{AC}\). Decide: are the rays different?
- Trial 1 (same ray): Place C on the same line as A and B, beyond B (so the order is A–B–C). Draw
Record your classification for each trial using one of these labels: same ray, opposite rays, or different rays.
Quick check questions
- In which trial(s) were A, B, and C collinear?
- In which trial was A between B and C?
- Which trial produced an angle at A that is not straight?
Translation Practice: Diagrams ↔ Statements
A. From a diagram to a statement
Use these sentence frames to describe what you see in a drawing:
- “Point ___ lies on ray ___.”
- “Rays ___ and ___ are opposite rays.”
- “Points ___, ___, and ___ are collinear.”
Example translations you should be able to write when appropriate:
- “B lies on ray AC” (meaning B is on the ray that starts at A and goes through C).
- “
\(\overrightarrow{AB}\)and\(\overrightarrow{AD}\)form an angle with vertex A.”
B. From a statement to a sketch
Sketch a small diagram for each statement. Use a point for the endpoint and an arrow for direction.
Statement 1: “B lies on ray AC.”
Sketch idea: draw ray
\(\overrightarrow{AC}\)starting at A; place B somewhere on the ray (possibly between A and C, or beyond C), making sure A, B, C are collinear and B is not on the opposite side of A.Statement 2: “
\(\overrightarrow{AB}\)and\(\overrightarrow{AC}\)are opposite rays.”Sketch idea: draw a straight line through A; put B on one side of A and C on the other side, then draw both rays starting at A in opposite directions.
Statement 3: “C does not lie on ray AB.”
Sketch idea: draw ray
\(\overrightarrow{AB}\); place C either off the line entirely or on the line but on the opposite side of A from B.
