The three siblings of the straight: a line reaches forever both ways, a ray starts at one point and runs on forever, a segment is finite — pinned at both ends.
The straight is the simplest shape there is — pull a string tight and you have it. But "straight" comes in three lengths of reach. One runs on forever in both directions. One has a starting point and shoots off forever in just one direction. And one is finite, fenced in by two endpoints. These are the line, the ray, and the segment — three siblings you will draw, name, and reason about for the rest of geometry. Along the way we meet two quiet facts that hold up every figure you will ever build: two points fix exactly one line, and the segment between two points is the shortest path — the very meaning of distance.
13.3.1 The three siblings, and how to name them
Take two points and draw the straight path through them. How far you let that path run gives you three different figures.
A line has no endpoints: it runs on forever in both directions. We draw arrowheads at both ends to say "this keeps going." A ray has one endpoint: it starts at a point and runs on forever in one direction — one arrowhead. A segment has two endpoints: it is finite, a clean piece with no arrows.
Naming matters, and the rules are simple:
Figure
Endpoints
How to name it
Notation
line
none
any two of its points, either order, or one lowercase letter
line AB = line BA = line l
ray
one
endpoint first, then any other point on it
ray OA (endpoint O)
segment
two
its two endpoints, either order
segment AB = segment BA
Watch out
For a ray, the endpoint must come first. Ray OA starts at O and goes toward A; ray AO starts at A and goes toward O — opposite directions, different rays. But for a line and a segment, order does not matter: line AB = line BA.
Toggle the same two points A and B between a line, a ray, and a segment. Watch the arrowheads and endpoints — and read off the correct notation.
Try it Line, ray, or segment?
Tap each one. Count the arrowheads (how far it reaches) and the filled dots (its endpoints).
Figure
Key idea
Arrowheads count the reach. Two arrows → a line (forever both ways). One arrow → a ray (forever one way). No arrows → a segment (finite). The filled dots are the endpoints: none, one, or two.
13.3.2 Two points determine exactly one line
Mark a single point on your paper. How many straight lines pass through it? You can spin a ruler around that point all day — infinitely many lines go through one point.
Now mark a second point. Try to draw a line through both. There is only one way to do it: exactly one line passes through two distinct points. That single line is sometimes written line AB.
Left: through one point A, infinitely many lines fan out. Right: through two points A and B, there is just one straight line.
Why it matters
This is why two nails and a taut string make a perfectly straight edge — and why a ruler "locks in" once you press it against two marked points. With one point the ruler still rocks; with two, it cannot. Two points pin a line.
13.3.3 The shortest path: a segment is the distance
Suppose an ant walks from A to B. It could wander along a curvy route, or take a couple of corners — but the shortest trip is always the straight one. Among all the paths joining two points, the straight segment is the shortest.
That shortest length has a name: it is the distance between the two points. So when we say "the distance from A to B," we mean exactly the length of segment AB.
From A to B: the straight segment (green) beats every bent or wiggly path (slate). Its length is the distance between the points.
Key idea
"A straight line is the shortest distance between two points." The segment joining them is that shortest path, and its length is the distance A B. We will measure that length carefully in Lesson 13.4.
13.3.4 Counting segments
Here is a question with a tidy answer. You have n points scattered on the page, with no three of them in a straight line. Join every pair with a segment. How many segments do you draw?
Pick a point. It connects to each of the other n − 1 points, giving n − 1 segments from that point. With n points that looks like n(n − 1) — but every segment got counted twice (once from each end), so we divide by 2:
number of segments = n(n − 1)2
Step n from 2 up to 6. Each new point joins to all the earlier ones, and the count climbs 1, 3, 6, 10, 15 — exactly n(n − 1) ÷ 2.
Try it How many segments?
Add points one at a time and watch the segments multiply. Can you predict the next count before you press +?
Points n3
A bridge back to counting
The same n(n − 1)2 counts the number of distinct lines through the points (no three on a line), and it is exactly the "choose 2 from n" you may meet again in algebra. Geometry and counting are the same idea wearing different clothes.
13.3.5 Positions of points and lines
Two figures sitting in the same plane can relate in only a few ways. First, a point and a line: the point is either on the line (the line passes right through it) or off it (it sits to one side).
Next, two distinct lines. There are exactly two possibilities. Either they cross — and two distinct lines can cross in at most one point — or they never meet, no matter how far you extend them. Lines that never meet are parallel, written with the symbol ∥: we say line a ∥ line b.
(1) point P lies on the line, point Q lies off it. (2) two lines cross at one point. (3) two parallel lines (∥) never meet.
Watch out
"Crossing" means meeting at a point. Two distinct straight lines can share at most one point — if they shared two, they would be the same line (remember 13.3.2: two points fix one line!). And parallel does not mean "pointing the same way for a while" — it means they never meet, however far you run them.
★ Recap
Figure
Endpoints
Reach
Notation
line
none
forever, both ways
line AB = BA, or line l
ray
one
forever, one way
ray OA (endpoint first)
segment
two
finite
segment AB = BA
And the facts that hold up every figure:
Two distinct points determine exactly one line.
The segment between two points is the shortest path; its length is the distance.
n points, no three on a line, give n(n − 1)2 segments.
Two distinct lines either cross at one point or are parallel (∥, never meet).
✎ Exercises 13.3
Name each figure in correct notation. (a) a straight path through points C and D with arrowheads at both ends; (b) a figure that starts at point S and runs forever through point T; (c) a finite piece with endpoints M and N.
Answer
(a) line CD (or line DC); (b) ray ST — endpoint S first; (c) segment MN (or segment NM).
Is ray OA the same as ray AO? Explain.
Answer
No. They have different endpoints and point in opposite directions: ray OA starts at O and heads toward A, while ray AO starts at A and heads toward O. (A line or segment would be the same either way — but a ray names its endpoint first.)
How many straight lines pass through two given points? Through a single point?
Answer
Through two distinct points: exactly one line. Through a single point: infinitely many. Two points pin a line down; one point lets it spin freely.
An ant must travel from A to B. Of a curved route, a two-corner route, and the straight route, which is shortest, and what is that shortest length called?
Answer
The straight route is shortest. Its length — the length of segment AB — is the distance between A and B.
You place 5 points on a page, no three on a line, and join every pair with a segment. How many segments do you draw? Show the step.
Answer
n(n − 1)2 = 5 × 42 = 202 = 10 segments.
Two distinct lines in a plane that never meet, however far you extend them, are called ___ . What symbol do we write, and what is the most points two distinct lines can share?
Answer
Parallel, written ∥ (for example a ∥ b). Two distinct lines can share at most one point — sharing two would make them the same line.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
The big idea
This lesson sorts the one word "straight" into three precise objects — line, ray, segment — and attaches the two foundational facts every later construction quietly assumes: two points determine a line, and the segment is the shortest path (the distance). The counting result n(n − 1)2 is a gentle, concrete preview of combinations.
Watch for this
The classic slip is ray OA = ray AO. Drill that a ray names its endpoint first, so swapping the letters reverses the direction — unlike a line or segment, where order is free. A second misconception: thinking lines must "look parallel for a stretch" to be parallel. Parallel is an all-or-nothing relationship — the lines simply never meet.
Common Core: this supports 4.G.A.1 (draw and identify points, lines, line segments, rays) and lays definitional groundwork for high-school G-CO.A.1 (precise definitions of line segment and ray) and the construction work in 7.G.