Lines and Angles · Lesson 1 of 9
Points, Segments, Lines, and Rays
“Discover the geometric objects that describe locations, straight paths, and directions.”
• Describe a point as a location and label it with a capital letter. • Distinguish a segment, a line, and a ray by their endpoints and extensions. • Name geometric objects correctly and explain why the order of ray labels matters. • Use diagrams to reason about how many lines pass through one or two distinct points.
From a Place to a Path
Imagine giving a friend directions to a tiny spot on a map. First you must identify the location. Then you can describe a path through it or a direction starting there. Geometry uses points, line segments, lines, and rays to express these ideas precisely. Their drawings are simple, but the ideas they represent are more exact than pencil strokes.
Touch a sharp pencil tip gently to paper. The dot helps you indicate a location. A real dot has some width, even if it is very small; the mathematical point represented by it has no length, breadth, or height. It tells us where something is, rather than how large it is. The tips of a needle and a compass also help us imagine a point.
An exact location with no length, breadth, or height. We represent it by a small dot and usually name it with a capital letter.
Place three dots far enough apart to distinguish them, and label them P, Z, and T. Read these names as “point P”, “point Z”, and “point T”. The letter is a label beside the dot; the dot indicates the location. Neither the shape nor the printed size of the letter is part of the point.
A Segment Is a Straight Connection
Choose two distinct points A and B. You could join them by a curved route, a zigzag, or a straight route. The straight connection is the shortest path between them. It begins at A, ends at B, and includes both endpoints. This is a line segment.
The straight path between two endpoints, including the endpoints. Segment AB and segment BA name the same segment.
A short bar above the endpoint letters is another way to indicate a segment. The bar represents its straight, finite connection. Reversing the endpoint order names the same connection.
Fold a sheet of paper and open it. A straight crease across the sheet models a segment because the visible crease ends at the edges of the paper. A drawn segment has a definite length, so it can be measured with a ruler. The actual pencil stroke has thickness, but the ideal segment has length without thickness.
Problem
A broken straight path joins L to M, M to P, P to Q, and Q to R. Name its four segments and identify points shared by two segments.
- 1.Name each straight piece by its endpoints: LM, MP, PQ, and QR.
- 2.M is shared by LM and MP. P is shared by MP and PQ. Q is shared by PQ and QR.
- 3.L belongs to only the first piece and R to only the last piece. Sharing an endpoint does not merge differently directed pieces into one segment.
A Line Continues in Both Directions
Now imagine extending segment AB past A and past B forever. The resulting straight object is a line. A sheet of paper can show only a small part of it, so arrowheads at both ends indicate that it continues beyond the drawing. A line has no endpoints and no finite total length to measure.
A straight object extending endlessly in both directions. A line through A and B can be named line AB, line BA, or by a small letter such as l.
To distinguish a line from a segment in symbolic writing, use a double-headed arrow above the two point labels. Its two arrowheads remind you of the endless extension in both directions.
Any two distinct points determine exactly one line. There is only one straight direction that passes through both locations. In contrast, if only one point is fixed, a line through it can be turned to many different directions. There is no last possible direction, so infinitely many lines pass through one point.
A Ray Has a Starting Point and a Direction
A ray keeps one endpoint and extends endlessly in one direction. The beam from a torch or lighthouse suggests a ray, although a real beam is not infinitely thin or endlessly visible. In a ray drawing, a dot or named point identifies the start and one arrowhead identifies the direction of continuation.
A part of a line that starts at one point and continues endlessly in one direction. In the name ray AP, A is the starting point and P lies in its direction.
A single-headed arrow above the point labels indicates a ray. Read the first letter as its starting point; the next letter identifies the direction.
A second point is used to show the direction. If O, B, and A occur in that order along the same ray, ray OA and ray OB name the same ray: both start at O and head in the same direction. Ray AO starts at A and heads toward O, so it is a different ray. Reversing labels is harmless for a segment or line, but changes a ray.
| Object | Endpoints | Endless extension | Naming order |
|---|---|---|---|
| Segment | Two | Neither direction | AB and BA are the same |
| Line | None | Both directions | AB and BA are the same |
| Ray | One starting point | One direction | AP and PA are different |
Problem
A ray starts at O, passes through B, and then through A. Are OA, OB, and AO names for the same ray?
- 1.Check the starting point first. OA and OB both start at O.
- 2.B and A are in the same direction from O, so OA and OB name the same ray.
- 3.AO starts at A and heads toward O. Its starting point and direction differ, so it is not the same ray.
Problem
A straight line l passes through E and F. Point D is drawn away from it. What does this tell you, and how would you draw intersecting lines through M?
- 1.E and F lie on l, so line EF is another name for l. D does not lie on l.
- 2.To draw two intersecting lines through M, draw one straight line through M and another in a different direction through M.
- 3.Put X and Y on the first line and P and Q on the second. Lines XY and PQ meet at M. Their arrowheads indicate continuation beyond the named points.
A short visible stroke with arrowheads can represent an endless line. Do not decide that it is a segment just because the drawing fits on a page. Use the endpoints, arrowheads, and description together.
Quiz
What does a mathematical point describe?
Which object has two endpoints?
How many different lines pass through two distinct points?
O, B, and A lie in that order on a ray starting at O. Which names describe the same ray?
Why can we not draw a complete line?
Which description correctly distinguishes a ray from a line?
Practice Problems
- Draw and label three points. Explain why the pencil dots are representations rather than the mathematical points themselves.
- Draw three routes from A to B, including a segment. Explain which is shortest.
- Draw a segment CD, a line EF, and a ray GH. Use appropriate endpoints and arrowheads.
- Draw several lines through one point. Explain why fixing a second distinct point changes the possibilities.
- A ray starts at P and passes through Q and R. State when PQ and PR name the same ray, and explain why QP is different.
- Draw a line containing E and F but not D. Add a ray starting at D and meeting the line at F.
- Draw a chain of four segments with five labelled points. Identify the points belonging to one piece and to two pieces.
- Draw a line with three points A, B, C on it. List three names for the line and three segments between the points.
Key Takeaways
• A point indicates a location and has no dimensions. • A segment is the shortest straight connection between two endpoints. • A line extends in both directions; a ray extends from one starting point in one direction. • Two distinct points determine one line, while one point lies on infinitely many lines. • The starting point must come first in a ray name.
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Angles as Turns