Three straight figures, three different reaches — and the two facts every construction leans on.
Point 4 of 5 in this lesson: 13.3.4 Counting segments
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.
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 |
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.
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.
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.
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.
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.
"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.
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
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.
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.
"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.
| 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:
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.
(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.
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?
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?
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.
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?
Parallel, written ∥ (for example a ∥ b). Two distinct lines can share at most one point — sharing two would make them the same line.
Six questions to lock it in. Tap the answer you think is right.
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.
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.