Points, Lines and Angles (Senior 1 Mathematics): free lessons | Ganzaa

Points, Lines and Angles

Lessons

Senior 1 · Mathematics

Points, Lines and Angles

5 lessons

Lesson 1 of 5

Points, lines, rays and segments

22 min

Boundary stones and straight edges

When a plot of land in Rwanda is registered, a surveyor places a boundary stone at each corner and records its exact position. Each stone marks a point, and each straight edge of the plot runs from one stone to the next. Geometry begins with exactly these ideas: a position, and the straight path between two positions. This unit builds everything else (angles, parallel lines and polygons) from them.

Definition

Point: an exact position. A point has no length, no width and no thickness; a dot on paper only shows where it is.

Notation: a point is named with a capital letter, for example "the point $A$ ".

Definition

Line: a straight path of points that goes on for ever in both directions. It has no endpoints and no width.

Notation: the line through the points $P$ and $Q$ is written $\overleftrightarrow{PQ}$ , or simply "line $P Q$ ". A line can also be named by one small letter, such as the line $l$ .

Definition

Ray (half-line): the part of a line that starts at one point, its endpoint, and goes on for ever in one direction.

Notation: $\overrightarrow{RS}$ is the ray that starts at $R$ and passes through $S$ . The endpoint is always written first.

Definition

Line segment: the part of a line between two endpoints, including the endpoints. A segment has a definite length.

Notation: $\overline{MN}$ is the segment with endpoints $M$ and $N$ ; its length is written $M N$ , for example $MN = 6\,\text{cm}$ .

A line has no endpoint, a ray has one, and a segment has two. Only the segment $\overline{MN}$ has a length you can measure.

Figure	Endpoints	Length	Symbol	Read as
line	none	none (it never ends)	$\overleftrightarrow{PQ}$	"line $P Q$ "
ray	one	none	$\overrightarrow{RS}$	"ray $R S$ "
segment	two	yes, e.g. $MN = 6\,\text{cm}$	$\overline{MN}$	"segment $M N$ "

Two facts that everything else rests on

Important

1. Through two different points there is exactly one straight line

Why: this is the first assumption (postulate) of Euclid's geometry, and you can see it with a string: pull a string tight between two nails and it always takes the same straight path. A surveyor uses this when she stretches a line between two boundary stones.

2. Two different lines meet in at most one point.

Why: if two lines shared two points, then by fact 1 they would be the same line.

Definition

Collinear points: points that lie on one straight line. Points that do not all lie on one line are non-collinear.

Example: three bus stops on a straight stretch of road are collinear.

Any two points are collinear. Three or more points may or may not be.

Definition

Plane: a flat surface that goes on for ever in every direction, like a perfectly flat, endless sheet of paper. All the figures in this unit are drawn in one plane.

How two lines in a plane can lie

Definition

Intersecting lines: lines that meet. The point where they meet is their point of intersection.

Parallel lines: lines in the same plane that never meet, however far they are extended. They stay the same distance apart.

Notation: $AB \parallel CD$ , read " $A B$ is parallel to $C D$ ". On a diagram, parallel lines carry matching arrow marks.

Perpendicular lines: lines that meet at a right angle (a quarter turn).

Notation: $PQ \perp RS$ , read " $P Q$ is perpendicular to $R S$ ". On a diagram, a small square marks the right angle.

The arrow marks say $AB \parallel CD$ ; the small square says $PQ \perp RS$ .

Worked example 1: naming segments and rays

Four collinear points: A, B, C and D.

(a) Name every line segment whose endpoints are two of the points $A, B, C, D$ .
(b) Name the two rays that start at $B$ .
(c) Is $\overrightarrow{AB}$ the same ray as $\overrightarrow{AC}$ ? Is $\overrightarrow{BA}$ the same ray as $\overrightarrow{BC}$ ?

Solution

Take the points in order and pair each with every point after it: $\overline{AB}, \overline{AC}, \overline{AD}, \overline{BC}, \overline{BD}, \overline{CD}$ . That is 6 segments. ( $\overline{BA}$ is the same segment as $\overline{AB}$ , so it is not counted again.)
From $B$ one ray goes left, through $A$ : $\overrightarrow{BA}$ . The other goes right, through $C$ and $D$ : $\overrightarrow{BC}$ , which can also be named $\overrightarrow{BD}$ .
$\overrightarrow{AB}$ and $\overrightarrow{AC}$ both start at $A$ and go right, so they are the same ray. $\overrightarrow{BA}$ and $\overrightarrow{BC}$ start at the same point but go in opposite directions, so they are different rays.

Worked example 2: how many cables?

Five sector offices are to be linked by straight fibre-optic cables, one cable for each pair of offices. No three offices lie on one straight line. How many cables are needed?

Five offices, ten cables: the 5 outer segments (black) and the 5 diagonals (blue).

Solution Each office is joined to the other 4, which gives $5 \times 4 = 20$ cable ends. Every cable has two ends, so each cable has been counted twice:

\text{number of cables} = \frac{5 \times 4}{2} = 10.

The same reasoning works for any number of points. For $n$ points, no three of them collinear, the number of segments joining them in pairs is

\frac{n(n-1)}{2}.

Worked example 3: parallel and perpendicular sides

A football pitch is a rectangle, 105 m long and 68 m wide. $M$ and $N$ are the middles of $\overline{AB}$ and $\overline{DC}$ , so $\overline{MN}$ is the halfway line.

The pitch ABCD and its halfway line MN. Right angles are marked at A and B.

(a) Name two pairs of parallel sides. (b) Name two lines perpendicular to $A B$ . (c) Is $M N$ parallel to $A D$ ?

Solution

Opposite sides of a rectangle are parallel: $AB \parallel DC$ and $AD \parallel BC$ .
Every corner of a rectangle is a right angle, so $AD \perp AB$ and $BC \perp AB$ .
Yes. $M N$ runs straight across the pitch from one long side to the other, like $A D$ and $B C$ , so $MN \parallel AD$ . In fact $MN \perp AB$ too.

Warning

Order matters for a ray, but not for a line or a segment

$\overline{AB}$ and $\overline{BA}$ are the same segment, and $\overleftrightarrow{AB}$ and $\overleftrightarrow{BA}$ are the same line.
$\overrightarrow{AB}$ and $\overrightarrow{BA}$ are different rays: the first starts at $A$ , the second starts at $B$ , and they point in opposite directions.
A line drawn in your book stops at the edge of the page, but the line itself does not. The arrowheads say "this goes on for ever"; leave them off only when you mean a segment.

Tip

Count the arrowheads: two for a line, one for a ray, none for a segment. The symbol over the letters tells the same story: $\overleftrightarrow{AB}$ , $\overrightarrow{AB}$ and $\overline{AB}$ .

Khan Academy: Intro to lines, line segments, and rays (4 min). Watch how each figure is named from two of its points, and why the ray's endpoint must be written first.

✦

Practice

Try every question before you open the answers. They get harder as you go.

Draw and label (a) the segment $\overline{KL}$ , (b) the ray $\overrightarrow{KL}$ , (c) the line $\overleftrightarrow{KL}$ . How many endpoints does each have?
True or false? (a) $\overline{AB}$ is the same as $\overline{BA}$ . (b) $\overrightarrow{AB}$ is the same as $\overrightarrow{BA}$ . (c) A ray can be measured with a ruler.
The points $P, Q$ and $R$ lie on a line in that order. Name (a) the three segments, (b) the two rays that start at $Q$ , (c) two different names for the ray that starts at $P$ and passes through $Q$ .
$K L M N$ is a rectangle. Name two pairs of parallel sides and two pairs of perpendicular sides.
How many lines can be drawn through (a) one point, (b) two different points?
Six towns, no three of them on one straight line, are each joined to every other town by a straight road. How many roads are there?
Four points are marked: $A, B$ and $C$ lie on one line, and $D$ does not. How many different lines pass through at least two of the four points?

Show answersHide answers

(a) 2 endpoints, $K$ and $L$ ; (b) 1 endpoint, $K$ , with an arrowhead beyond $L$ ; (c) no endpoints, arrowheads at both ends.
(a) True. (b) False: they start at different points and point in opposite directions. (c) False: a ray goes on for ever, so it has no length.
(a) $\overline{PQ}, \overline{QR}, \overline{PR}$ . (b) $\overrightarrow{QP}$ and $\overrightarrow{QR}$ . (c) $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ .
$KL \parallel NM$ and $LM \parallel KN$ . Perpendicular pairs include $KL \perp LM$ and $LM \perp MN$ (any two sides that meet at a corner).
(a) Infinitely many. (b) Exactly one.
$\frac{6 \times 5}{2} = 15$ roads.
4 lines: the line through $A, B$ and $C$ , plus $D A$ , $D B$ and $D C$ . (The formula $\frac{4 \times 3}{2} = 6$ does not apply, because three of the points are collinear: $A B$ , $A C$ and $B C$ are one line.)

Sources and credits

10.1: Points, Lines, and Planes, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
Video: Intro to lines, line segments, and rays, Khan Academy, embedded from YouTube.

The Rwandan examples, the diagrams, the counting rule for segments, the practice items and the quiz questions were written by Ganzaa.

1Boundary stones and straight edges

2Two facts that everything else rests on

3How two lines in a plane can lie

Boundary stones and straight edges

Two facts that everything else rests on

How two lines in a plane can lie