What you'll learn
An angle is an amount of turn, measured in degrees. A quarter turn is 90° and is called a right angle; a half turn is 180°, which is a straight line; a full turn all the way round is 360°. A shape has a line of symmetry if you could fold it along that line and both halves would match exactly, and a regular shape has exactly as many lines of symmetry as it has sides — a square has 4, a regular hexagon has 6. This topic covers naming angles as acute, right, obtuse or reflex, turning clockwise and anticlockwise, using the compass points, the angle facts on a straight line and inside a triangle, and counting lines of symmetry. Geometry carries 6 of the 40 items on the SEA Mathematics paper, and angle-and-symmetry questions make up most of that block.
Key terms and definitions
Angle — the amount of turn between two lines that meet at a point, measured in degrees (°).
Degree (°) — the unit of angle. There are 360 degrees in a full turn.
Right angle — a quarter turn, 90°. The square corner of a book or a tile.
Acute angle — an angle smaller than 90°.
Obtuse angle — an angle bigger than 90° but smaller than 180°.
Straight angle — a half turn, exactly 180°.
Reflex angle — an angle bigger than 180° but smaller than 360°.
Clockwise — turning the way the hands of a clock go: from twelve towards three.
Anticlockwise — turning the other way, from twelve towards nine.
Perpendicular — two lines that meet at a right angle.
Parallel — two lines that run alongside each other and never meet.
Line of symmetry — a fold line that makes the two halves of a shape match exactly. Also called a mirror line.
Core concepts
Angles are turns
Stand facing the front of the classroom. Turn to face the side wall and you have made a quarter turn, 90°. Turn again to face the back and you have made a half turn, 180°. Twice more and you are back where you began, a full turn of 360°.
So a quarter turn is 90°, a half turn 180°, a three-quarter turn 270° and a full turn 360°.
Because a full turn is 360°, one eighth of a turn is 360 ÷ 8 = 45°, and one twelfth of a turn is 360 ÷ 12 = 30°. That last one is why each hour mark on a clock is 30° apart: the hour hand moving from twelve to three passes three marks, which is 3 × 30 = 90°, a right angle.
Naming an angle by its size
| Name | Size |
|---|---|
| Acute | less than 90° |
| Right | exactly 90° |
| Obtuse | more than 90°, less than 180° |
| Straight | exactly 180° |
| Reflex | more than 180°, less than 360° |
Learn these in order, because the paper often asks only for the name. An angle of 37° is acute; 128° is obtuse; 200° is reflex.
To judge an angle without measuring, compare it with the corner of a piece of paper. Narrower than the paper corner is acute; wider is obtuse.
Clockwise, anticlockwise and the compass
The four main compass points, going clockwise, are North, East, South, West. A useful order to say aloud is "Never Eat Sour Watermelon".
Each step from one to the next is a quarter turn, 90°. So:
- From North, a quarter turn clockwise faces East.
- From North, a half turn faces South, whichever way you turn.
- From North, a three-quarter turn clockwise faces West.
- From North, a quarter turn anticlockwise also faces West.
The in-between points are North-East, South-East, South-West and North-West, each 45° from its neighbours.
The trap is the direction. A quarter turn clockwise and a quarter turn anticlockwise from the same start land on opposite sides, so always read the word clockwise or anticlockwise first.
Angles on a straight line and at a point
Two facts are worth memorising because they turn a hard-looking item into one subtraction.
Angles on a straight line add up to 180°. If one angle is 115°, the angle beside it on the same straight line is 180 − 115 = 65°.
Angles round a point add up to 360°. If three angles meeting at a point are 90°, 130° and one unknown, the unknown is 360 − 90 − 130 = 140°.
Angles in a triangle
The three angles inside any triangle add up to 180°.
So if two angles of a triangle are 65° and 48°, the third is 180 − 65 − 48. Take it in two steps: 65 + 48 = 113, and 180 − 113 = 67°.
This also explains an equilateral triangle. Its three angles are equal, and 180 ÷ 3 = 60°, so every angle in an equilateral triangle is 60°.
Lines of symmetry
A shape is symmetrical if it can be folded so that one half lies exactly on top of the other. The fold line is the line of symmetry, and a shape may have several, one, or none at all.
The most useful rule is this: a regular polygon has as many lines of symmetry as it has sides.
| Shape | Lines of symmetry |
|---|---|
| Equilateral triangle (regular, 3 sides) | 3 |
| Square (regular, 4 sides) | 4 |
| Regular pentagon (5 sides) | 5 |
| Regular hexagon (6 sides) | 6 |
| Regular octagon (8 sides) | 8 |
| Circle | too many to count |
Shapes that are not regular are the ones that catch people out:
| Shape | Lines of symmetry |
|---|---|
| Rectangle (not a square) | 2 |
| Rhombus (not a square) | 2 |
| Isosceles triangle | 1 |
| Kite | 1 |
| Parallelogram (not a rhombus) | 0 |
| Scalene triangle | 0 |
| Trapezium (not isosceles) | 0 |
The rectangle is the famous mistake. It looks as though the diagonals should be fold lines, but fold a long rectangle along a diagonal and the corners do not meet. Its only fold lines are the one across its middle from side to side and the one from top to bottom, giving 2.
The parallelogram has no line of symmetry at all, even though it looks balanced, because it is a pushed-over rectangle and no fold makes the halves match.
Worked examples
Example 1: Angles on a straight line
Two angles sit side by side on a straight line. One of them measures 115°. What is the size of the other, and what kind of angle is it?
Angles on a straight line add to 180°.
The other angle = 180 − 115.
Take it in steps: 180 − 100 = 80, then 80 − 15 = 65°.
Since 65 is less than 90, it is an acute angle.
Check by adding back: 115 + 65 = 180. Correct.
The first angle, 115°, is obtuse. Whenever two angles make a straight line, one is acute and the other obtuse, unless both are exactly 90°.
Example 2: Turning on the compass
Kern is facing North. He makes a three-quarter turn clockwise, and then a further quarter turn clockwise. Which direction is he facing at the end, and how many degrees has he turned altogether?
Take it one step at a time. Going clockwise the order is North, East, South, West, and back to North.
A three-quarter turn is three quarter-turns: North to East, East to South, South to West. He is now facing West, having turned 3 × 90 = 270°.
A further quarter turn clockwise takes him from West back to North.
Total turn: 270 + 90 = 360°, which is a full turn, and that is why he has ended up facing the way he started.
Had the second turn been anticlockwise, he would have gone from West back to South instead.
Example 3: Counting lines of symmetry
Shanice cuts out five shapes: a square, a regular hexagon, a rectangle, an isosceles triangle and a parallelogram. She folds each one to find its lines of symmetry. How many lines does she find in total, and which shape has none?
Take each in turn. The square and the hexagon are regular, so each has as many lines as it has sides: 4 and 6.
The rectangle has 2 — one fold across the middle from left to right, and one from top to bottom. The diagonals do not work.
The isosceles triangle has 1, the fold running from the point at the top straight down to the middle of the bottom side.
The parallelogram has 0.
Total: 4 + 6 + 2 + 1 + 0. Add in steps: 4 + 6 = 10, 10 + 2 = 12, 12 + 1 = 13 lines of symmetry, and the parallelogram is the shape with none.
Common mistakes and how to avoid them
Giving a rectangle 4 lines of symmetry. Fix: only a square has 4. A rectangle has 2, because folding along a diagonal does not make the halves match.
Giving a parallelogram a line of symmetry. Fix: it has none. Being balanced-looking is not the same as being symmetrical.
Mixing up acute and obtuse. Fix: acute is the small, sharp one — think of a cute little angle. Obtuse is the wide one.
Forgetting the reflex angle. Answering "180°" for anything bigger than a straight line. Fix: anything between 180° and 360° is reflex.
Turning the wrong way. Fix: underline clockwise or anticlockwise in the question before you start turning.
Adding turns wrongly. Thinking a three-quarter turn is 180°. Fix: three quarters of 360 is 3 × 90 = 270°.
Forgetting that a triangle's angles make 180°. Fix: subtract both known angles from 180, and check by adding all three back.
Confusing parallel and perpendicular. Fix: parallel lines never meet; perpendicular lines meet at a right angle.
How parents can help at home
Angles and symmetry are the easiest topics to practise without a book, because both are about looking.
Hunt for right angles. Ask your child to find five right angles in the room — a window frame, a door, a tile. Then ask for one angle bigger than a right angle and one smaller, and for the name of each.
Use the corner of a page. A folded piece of paper makes a perfect right-angle tester. Hold it against a corner and ask, "is this acute, right or obtuse?"
Turn while walking. On the way to the shop, say "make a quarter turn to your left" and ask what that is in degrees. Doing this with the body fixes it better than a page does.
Fold things in half. Give your child a leaf, a napkin or a cut-out heart and ask them to fold it so the halves match. Then ask whether it can be folded a different way too. That question is exactly what "how many lines of symmetry" means.
When your child is stuck on symmetry, the prompt is "could you fold it there and would the halves land on top of each other?" — and if they are unsure, actually fold it. When they are stuck on a turn, the prompt is "stand up and do it".
You do not need to remember the angle facts yourself. Ask your child to check their answer by adding: angles on a straight line must total 180, angles inside a triangle must total 180, and angles round a point must total 360. If the total is wrong, so is the answer.
Exam technique for angles, turn and symmetry
Learn the four turn sizes cold: 90°, 180°, 270°, 360°. Many items are just these numbers dressed as compass directions.
Read the direction word before doing anything. Clockwise and anticlockwise give different answers from the same start.
When an item names an angle, decide the family first — acute, right, obtuse or reflex — because that alone rules out most of the answer choices.
Use the straight line and triangle facts as subtractions, and check by adding your answer back to the given ones.
For symmetry, ask whether the shape is regular. If it is, the answer is the number of sides. If it is not, work through the folds one at a time and remember that a rectangle has 2 and a parallelogram has 0.
Write the degree sign. An answer of "65" where "65°" was wanted can lose the mark.
Do not linger on one Geometry item. There are only 6 among the 40, and Number carries far more of the paper.
Quick revision summary
- A full turn is 360°, a half turn 180°, a quarter turn (right angle) 90°, a three-quarter turn 270°.
- Acute is under 90°, right is exactly 90°, obtuse is between 90° and 180°, reflex is between 180° and 360°.
- Compass points clockwise: North, East, South, West, each a quarter turn apart.
- Angles on a straight line add to 180°; angles round a point add to 360°.
- The angles inside any triangle add to 180°; each angle of an equilateral triangle is 60°.
- A line of symmetry is a fold that makes the halves match exactly.
- A regular polygon has as many lines of symmetry as sides: square 4, regular hexagon 6, regular octagon 8.
- Rectangle 2, rhombus 2, isosceles triangle 1, kite 1, parallelogram 0, scalene triangle 0.
- Parallel lines never meet; perpendicular lines meet at a right angle.