Points, line segments, rays and lines
The tip of a sharp pencil marks a point. A stretched rubber band is like a line segment. A torch beam is like a ray, because it starts at the torch and keeps going. A road drawn on a map that never ends is like a line.
A point shows an exact place and has no size. A line segment is the straight path between two points and has two end points. A ray starts at one point and goes on for ever in one direction. A line goes on for ever in both directions and has no end points.
In the figure: A point P, segment AB, ray AB (starts at A, passes B, never ends) and line AB (arrows on both sides).
Rule 1: Count the end points — A line segment has 2 end points, a ray has 1 and a line has 0. Only a line segment has a length that can be measured.
Points, line segments, rays and lines
In the app you can move this figure with your finger.
Open in the app- Easy: A scale line drawn between dots P and Q is segment PQ: 2 end points, so it can be measured.
- Medium: Sunlight from a lamp goes out in one direction from the bulb: it is a ray with 1 end point.
- Exam-type: Which of a segment, a ray and a line can you measure with a ruler? Only the segment, because the ray and the line never end.
Rule 2: Counting segments from points — Exactly one line passes through two points, but many lines pass through one point. To count segments, join every pair of points once. With n points, no three in a line, the count is (n − 1) + (n − 2) + … + 1.
- Easy: 3 points A, B, C not in a line: segments AB, BC, CA = 3.
- Medium: 4 points, no three in a line: 3 + 2 + 1 = 6 segments.
- Exam-type: 6 points, no three in a line: 5 + 4 + 3 + 2 + 1 = 15 segments.
Ray AB starts at A and passes through B. Ray BA starts at B. They are different rays.
Calling a ray and a line the same, or thinking a ray AB and a ray BA are equal. The starting point is always written first.
End points: segment 2, ray 1, line 0. Memory clue: Remember “2-1-0”: Stops at both ends, runs one way, never stops.
Spot the difference
| Property | Line segment | Ray | Line |
|---|---|---|---|
| End points | 2 | 1 | 0 |
| Length | can be measured | cannot be measured | cannot be measured |
| Goes on for ever | no | one way | both ways |
| Name | AB | ray AB | line AB |
Practice questions
- 1How many end points does a line segment have?
- 2How many end points does a ray have?
- 3Which of segment, ray and line has no end points?
- 4How many lines pass through two given points?
- 55 points, no three in a line, are joined in all possible pairs. How many segments are formed?
- 6What can be measured with a ruler: ray AB or segment AB?
- 74 points A, B, C, D lie on one line in this order. How many different segments can be named?
- 8Are ray AB and ray BA the same? Give a reason.
Show answers (try them yourself first)
- 11) 2
- 22) 1
- 33) the line
- 44) exactly one
- 55) 10 (4 + 3 + 2 + 1)
- 66) segment AB, because it has two end points
- 77) 6: AB, AC, AD, BC, BD, CD
- 88) No. Ray AB starts at A and ray BA starts at B, so they go in different directions.
Angles: arms, vertex and comparing
A door opened a little and then wide, a pair of scissors, the two hands of a clock and an open book all show angles. The more the door turns, the bigger the angle.
An angle is formed when two rays start from the same point. The two rays are the arms and the common starting point is the vertex. The size of an angle is the amount of turn between its arms. We write an angle as ∠AOB with the vertex O in the middle.
In the figure: ∠AOB: vertex O, arms OA and OB. The orange arc shows the turn between the arms.
Rule 1: Name an angle correctly — The vertex letter always goes in the middle. ∠PQR and ∠RQP are the same angle. When many angles share a vertex, one letter is not enough, so use three letters.
- Easy: Arms OA and OB give ∠AOB, vertex O.
- Medium: In ∠DBE the vertex is B, the middle letter.
- Exam-type: Three points A, B, C not in a line: at each point choose the other two. Angles ∠ABC, ∠BCA, ∠CAB = 3. With 4 points: 4 × 3 = 12.
Rule 2: Compare angles by turning — Place one angle over the other with the vertices together and one arm on the other. The angle whose second arm falls outside is bigger. If both arms fit exactly, the angles are equal. The length of the arms never matters.
Angles: arms, vertex and comparing
In the app you can move this figure with your finger.
Open in the app- Easy: 40° and 70°: the 70° angle opens wider, so it is bigger.
- Medium: Scissors opened wide make a bigger angle than scissors opened a little, even though the blades are the same.
- Exam-type: Angle X has 10 cm arms and turns 30°. Angle Y has 5 cm arms and turns 45°. Y is bigger because 45° > 30°; arm length is ignored.
The size of an angle is the amount of turn, not how long the arms are drawn.
Thinking that a drawing with longer arms has a bigger angle.
Vertex in the middle; compare the opening, not the arm length. Memory clue: “Open Wide = Big Angle”: think of a mouth, not of a long tongue.
Spot the difference
| Part | What it is | Example | Note |
|---|---|---|---|
| Arm | a ray of the angle | OA, OB | two arms |
| Vertex | common starting point | O | middle letter |
| Angle | turn between arms | ∠AOB | same as ∠BOA |
Practice questions
- 1Two rays with a common starting point form what?
- 2Which letter is the vertex of ∠DBE?
- 3If the arms of an angle are drawn longer, what happens to the angle?
- 4Another name for ∠PQR?
- 5How many angles can be named from 4 points, no three in a line?
- 6Where is the vertex of the angle made by the two hands of a clock?
- 7How many angles can be named from 5 points, no three in a line?
- 8A friend says “∠ABC is bigger because its arms are longer.” Is the friend right?
Show answers (try them yourself first)
- 11) an angle
- 22) B
- 33) it stays the same
- 44) ∠RQP
- 55) 12 (4 × 3)
- 66) at the centre of the clock
- 77) 30 (5 × 6)
- 88) No. The size depends on the turn, not on the arm length.
Types of angles and perpendicular lines
The corner of a book is a right angle. A half-open door is an acute angle. A fully opened door that lies flat is a straight angle. A full turn of a wheel brings it back to where it started.
A right angle is 90°, a quarter of a full turn. A straight angle is 180°, half a turn. A full turn is 360°. An acute angle is less than 90°, an obtuse angle is between 90° and 180°, and a reflex angle is between 180° and 360°. Two lines that meet at a right angle are perpendicular.
In the figure: The five kinds of angle, from small to large.
Rule 1: Classify by the number of degrees — Compare the angle with 90°, 180° and 360°. Less than 90° is acute; exactly 90° is right; more than 90° but less than 180° is obtuse; exactly 180° is straight; more than 180° but less than 360° is reflex.
Types of angles and perpendicular lines
In the app you can move this figure with your finger.
Open in the app- Easy: 35° is less than 90°: acute.
- Medium: 120° is between 90° and 180°: obtuse.
- Exam-type: 270° is more than 180° and less than 360°: reflex. It is also 3 quarter turns (3 × 90°).
Rule 2: Fractions of a turn give angles — A full turn is 360°. Find an angle by dividing 360° or taking a part: half = 180°, quarter = 90°, one eighth = 45°. Two right angles together make a straight angle.
- Easy: Half of a straight angle: 180° ÷ 2 = 90°, a right angle.
- Medium: One eighth of a full turn: 360° ÷ 8 = 45°, an acute angle.
- Exam-type: How many right angles in a full turn? 360° ÷ 90° = 4.
Right 90°, straight 180°, full turn 360°. Perpendicular lines meet at 90°.
Calling 180° an obtuse angle. 180° is a straight angle; obtuse is only strictly between 90° and 180°.
Check the boundary numbers 90°, 180°, 360° first. Memory clue: Acute, Right, Obtuse, Straight, Reflex: say the five names in this order and the angles grow like a growing child.
Spot the difference
| Angle | Measure | Example | Quick test |
|---|---|---|---|
| Acute | less than 90° | 45° | smaller than the corner of a book |
| Right | 90° | 90° | exactly the corner of a book |
| Obtuse | between 90° and 180° | 120° | wider than the corner of a book |
| Straight | 180° | 180° | a straight line |
| Reflex | between 180° and 360° | 270° | more than a straight line |
Practice questions
- 1How many degrees in a right angle?
- 2How many degrees in a straight angle?
- 3An angle less than 90° is called what?
- 4What type of angle is 120°?
- 5A quarter of a full turn is how many degrees?
- 6How many right angles are in one straight angle?
- 7Which is the largest whole-number angle that is still acute?
- 8One eighth of a full turn is how many degrees and which type?
Show answers (try them yourself first)
- 11) 90°
- 22) 180°
- 33) acute angle
- 44) obtuse
- 55) 90°
- 66) 2 (180° ÷ 90°)
- 77) 89°
- 88) 45°, acute (360° ÷ 8)
Measuring angles and the protractor
A pizza cut into 8 equal slices has 45° at the centre of each slice. A clock face is a full turn split into 12 equal angles of 30°. A protractor is a ruler for angles.
The unit of angle is the degree (°). One full turn is divided into 360 equal parts and each part is 1°. A protractor measures angles in degrees. Put its centre on the vertex and one arm on 0°, then read where the other arm crosses the same scale.
In the figure: A protractor: the first arm is at 20° and the second at 110°. The angle between them is 110° − 20° = 90°.
Rule 1: Read the same scale — A protractor has two scales. Use the scale that starts at 0° on your base arm. If neither arm is on 0°, subtract the two readings on the same scale.
Measuring angles and the protractor
In the app you can move this figure with your finger.
Open in the app- Easy: One arm at 0° and the other at 70°: the angle is 70°.
- Medium: Arms at 20° and 55°: 55° − 20° = 35°.
- Exam-type: Arms at 40° and 125°: 125° − 40° = 85°, an acute angle (less than 90°).
Rule 2: Make angles by folding and dividing — Folding an angle in half halves it: 180° → 90° → 45° → 22.5°. To split a full turn into equal angles, divide 360° by the number of parts.
- Easy: Fold a right angle in half: 90° ÷ 2 = 45°.
- Medium: Split a circle into 10 equal angles: 360° ÷ 10 = 36° each.
- Exam-type: Split a circle into 9 equal angles: 360° ÷ 9 = 40° each; two of them together make 80°.
One degree is 1/360 of a full turn. Always read the scale that starts at 0° on your arm.
Reading the wrong scale of the protractor, which gives 180° minus the right answer.
Two readings on the same scale: big minus small. Equal parts of a turn: 360° ÷ parts. Memory clue: “Start at Zero, Same Scale, Subtract”.
Spot the difference
| Splitting a full turn | Number of parts | Calculation | Each angle |
|---|---|---|---|
| Quarter | 4 | 360° ÷ 4 | 90° |
| Clock hours | 12 | 360° ÷ 12 | 30° |
| Pentagon centre | 5 | 360° ÷ 5 | 72° |
| Decagon centre | 10 | 360° ÷ 10 | 36° |
Practice questions
- 1How many equal parts is a full turn divided into for degrees?
- 2What is a quarter of 360°?
- 3A right angle is folded in half. What is each part?
- 4Arms at 20° and 55° on the same scale. Angle?
- 5A circle is split into 12 equal angles. Each angle?
- 6A 45° angle is folded in half. Each part?
- 7Arms at 40° and 125° on the same scale. Angle?
- 8A circle is split into 9 equal angles. What do two of them add to?
Show answers (try them yourself first)
- 11) 360
- 22) 90°
- 33) 45°
- 44) 35°
- 55) 30°
- 66) 22.5°
- 77) 85°
- 88) 80° (each is 360° ÷ 9 = 40°)
Drawing angles, bisectors and angle sums
Every hour the hour hand of a clock turns 30°. A kite maker cuts a triangle of cloth and knows its three corners always add up to the same total. A line that cuts a corner into two equal halves is called a bisector.
To draw an angle, draw the base arm, put the protractor centre on the vertex, mark the angle on the scale and join. A bisector splits an angle into two equal angles. The three angles of any triangle add up to 180°.
In the figure: A triangle with angles 50°, 60° and 70°. They add up to 180°.
Rule 1: Clock angles — A clock has 12 hours in 360°, so each hour is 30°. At an exact hour the angle is hours × 30°. If that is more than 180°, subtract it from 360° to get the smaller angle.
Drawing angles, bisectors and angle sums
In the app you can move this figure with your finger.
Open in the app- Easy: At 2 o’clock: 2 × 30° = 60°.
- Medium: At 5 o’clock: 5 × 30° = 150°.
- Exam-type: At 8 o’clock: 8 × 30° = 240° is more than 180°, so the smaller angle is 360° − 240° = 120°.
Rule 2: Bisector and triangle sum — A bisector halves the angle: divide by 2. In a triangle the three angles total 180°, so the third angle = 180° − (the other two).
Drawing angles, bisectors and angle sums
In the app you can move this figure with your finger.
Open in the app- Easy: Bisect 60°: 60° ÷ 2 = 30° and 30°.
- Medium: Triangle with angles 50° and 60°: third angle = 180° − 110° = 70°.
- Exam-type: One angle is 100° and the other two are equal: (180° − 100°) ÷ 2 = 40° each.
Rule 3: Three angles of a triangle — Move the sliders: whichever two angles you choose, the third is fixed by the total 180°.
Drawing angles, bisectors and angle sums
In the app you can move this figure with your finger.
Open in the app- Easy: All three equal: 180° ÷ 3 = 60° each.
- Medium: A right angle and 35°: third = 180° − (90° + 35°) = 55°.
- Exam-type: Two angles of 40° and 40°: third = 180° − 80° = 100°, an obtuse angle.
Triangle angles total 180°. Each hour on a clock is 30°.
Forgetting that the clock angle can be measured two ways and giving the bigger one. Give the smaller angle unless asked otherwise.
Clock: hours × 30°, then take the smaller of that angle and (360° − that angle). Triangle: 180° minus the two known angles. Memory clue: “30 for every hour, 180 for every triangle”.
Spot the difference
| Time | Hour hand turn | Other angle | Smaller angle |
|---|---|---|---|
| 3 o’clock | 90° | 270° | 90° |
| 4 o’clock | 120° | 240° | 120° |
| 7 o’clock | 210° | 150° | 150° |
| 9 o’clock | 270° | 90° | 90° |
Practice questions
- 1Angle between the clock hands at 3 o’clock?
- 2Angle between the clock hands at 6 o’clock?
- 3Sum of the three angles of a triangle?
- 4Bisect an 84° angle. Each half?
- 5Two angles of a triangle are 50° and 60°. Third?
- 6Angle between the clock hands at 4 o’clock?
- 7Smaller angle between the hands at 7 o’clock?
- 8One angle is 100° and the other two are equal. Each of them?
Show answers (try them yourself first)
- 11) 90°
- 22) 180°
- 33) 180°
- 44) 42°
- 55) 70°
- 66) 120°
- 77) 150° (210° one way, 360° − 210° = 150° the other)
- 88) 40° each (180° − 100° = 80°, 80° ÷ 2)