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Angle properties
Angles on a straight line, around a point, and vertically opposite angles.
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Angles are everywhere: a door opening, a slice of pizza, the hands of a clock, a ramp at the skate park. In this lesson you will learn how to:
- say what an angle is, and measure turns in degrees
- name acute, right, obtuse, straight and reflex angles, and estimate their size
- find missing angles on a straight line (they add up to 180°)
- find missing angles around a point (they add up to 360°)
- use vertically opposite angles, where two lines cross
- solve angle problems, and give a reason for each step
Dark blue: whole numbers
Red dot: the decimal point
Blue: digits after the point
1. What is an angle?
Key idea
An angle is an amount of turn. Two arms meet at a point. The more one arm turns away from
the other, the bigger the angle. We measure angles in degrees, written with a small circle: 90°
means 90 degrees. One full turn is 360°.
Turns and degrees
Quarter turn
90°: a right angle
Half turn
180°
Three-quarter turn
270°
Full turn
360°
A quarter turn is a right angle, like the corner of a book. We mark it with a small square instead of a curve. Two right angles make a straight line, and four make a full turn: 4 × 90 = 360.
Which angle is bigger?
Neither! Both arms open by the same amount, so the angles are the same size. The size of an angle is how far it turns. How long the arms are does not matter.
Watch out
Long arms do not make a bigger angle. If you drew the arms of angle B longer, they would land right on top of
angle A.
The angle maker
Drag the red dot to open or close the angle. Watch its name change as it grows.
How to play
- Drag the red dot round, or use the buttons (or the arrow keys) to make the angle bigger or smaller, one degree or ten at a time. The strip underneath shows which kind of angle you have made.
- Then try the challenge and press Check. Two with no slips moves you up a level. Level 1 asks for a kind of angle, level 2 for an exact number of degrees, and level 3 for turns and puzzles.
2. Acute, right, obtuse and reflex
Key idea
Every angle has a name that tells you roughly how big it is. Compare it with a right angle (90°)
and a straight line (180°): is it smaller, the same, or bigger?
Acute
less than 90°
Right angle
exactly 90°
Obtuse
between 90° and 180°
Straight angle
exactly 180°
Reflex
between 180° and 360°
Full turn
exactly 360°
Example 1: what kind of angle is 135°?
- Is it bigger than a right angle?135 is more than 90: yes
- Is it smaller than a straight line?135 is less than 180: yes
AnswerAn obtuse angle.
Example 2: estimate the size of this angle
- It is smaller than a right angle, so it is acute:less than 90°
- It is a little less than half a right angle:half of 90 is 45
AnswerAbout 40°.
Watch out
Look for the curve. Two arms make two angles: the one inside and the big one the other way round. The curve (or
the square) shows which one the question means. If it goes the long way round, it is a reflex angle.
Name that angle
How to play
- Look at the curve, then tap the name of the angle it marks. The angle can point any way round.
- Two in a row with no slips moves you up a level. Level 1 has acute, right and obtuse angles, level 2 adds straight and reflex angles, and level 3 asks you to estimate how many degrees.
3. Angles on a straight line
Key idea
A straight line is a half turn, so the angles on a straight line, at one point, add up to 180°.
To find a missing angle, add up the angles you know, then take that away from 180.
Example 1: find the missing angle
- Angles on a straight line add up to 180°:180 − 130 = 50
Answer50°
Example 2: three angles on a line
- Add the angles you know:45 + 65 = 110
- Take that away from 180:180 − 110 = 70
Answer70°
Watch out
Check that your answer looks right. In Example 1 the missing angle is acute, and 50° is acute. If you
get an obtuse answer for an angle that is clearly acute, look again.
Split the line
Drag the red dot. However you split the line, the two angles always add up to 180°.
How to play
- Play with the line first. Then find the missing angle underneath: type each step and press Check.
- Two with no slips moves you up a level. Level 1 has two angles in tens, level 2 any two or three angles, and level 3 has equal angles marked with the same letter.
4. Angles around a point
Key idea
Going all the way round a point is a full turn, so the angles around a point add up to 360°.
To find a missing angle, add up the angles you know, then take that away from 360.
Example 1: find the missing angle
- The square means a right angle. Add the angles you know:90 + 150 = 240
- Angles around a point add up to 360°:360 − 240 = 120
Answer120°
Example 2: the angle the other way round
- The two angles go all the way round the point:360 − 110 = 250
Answer250°, a reflex angle
Example 3: three equal angles, each marked a
- The same letter means the same size. Share 360° equally:360 ÷ 3 = 120
Answera = 120°
Watch out
180 is for a straight line and 360 is for all the way round a point. Before you take away, ask yourself:
does this go half way round, or all the way?
Around the point
Drag any red dot to turn that arm. The angles change, but they always add up to 360°.
How to play
- Play with the arms first. Then find the missing angle underneath, one step at a time.
- Two with no slips moves you up a level. Level 1 has three angles, level 2 has four, and level 3 has reflex angles and equal angles marked with the same letter.
5. Vertically opposite angles
Key idea
When two straight lines cross, they make four angles, like an open pair of scissors. The angles
opposite each other are equal. They are called vertically opposite angles. Angles next
to each other are on a straight line, so they add up to 180°.
Example: two lines cross. Find a, b and c.
- b is opposite the 40°. Vertically opposite angles are equal:b = 40
- a is next to the 40°, on a straight line:a = 180 − 40 = 140
- c is opposite a:c = 140
- Check: all four go round the point:40 + 140 + 40 + 140 = 360
Answera = 140°, b = 40°, c = 140°
Watch out
“Vertically” does not mean up and down. It comes from vertex, the point where the lines meet.
Vertically opposite angles can point any way: left and right, up and down, or slanting.
The scissors
Drag the red dot to open and close the scissors. The angles opposite each other stay the same colour, and the same size.
How to play
- Open and close the scissors first. Then find the missing angle underneath.
- Two with no slips moves you up a level. Level 1 asks for the opposite angle, level 2 for the angle next door, and level 3 for all three.
6. Angle detective
Key idea
Bigger angle problems are a chain of small steps. At each step, use one fact and say which one: this is your
reason.
- Angles on a straight line add up to 180°.
- Angles around a point add up to 360°.
- Vertically opposite angles are equal.
Example 1: three lines cross at one point. Find a and b.
- a, 40° and 75° are on a straight line:a = 180 − 40 − 75 = 65
- b is vertically opposite a:b = 65
Answera = 65° and b = 65°
Example 2: the two angles marked a are equal. Find a.
- Add the angles you know:140 + 100 = 240
- Around a point is 360°, so the two a’s make:360 − 240 = 120
- Share between the two equal angles:120 ÷ 2 = 60
Answera = 60°
Example 3: what is the smaller angle between the hands of a clock at 4 o’clock?
- The 12 numbers split a full turn into 12 equal parts:360 ÷ 12 = 30
- From the 12 to the 4 is 4 parts:4 × 30 = 120
Answer120°
Watch out
Only use a fact when it really fits. Angles add up to 180° only if they sit together on one straight
line at one point. If the line bends, or the angles are at different points, the fact does not
work.
Angle detective
How it works
- For each step, first pick the reason, then type the angle.
- Two with no slips moves you up a level. Level 1 is straight lines, level 2 is around a point, level 3 is crossing lines, level 4 needs two facts, one after the other, and level 5 has equal angles and clock hands.
7. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Move up when you are getting most of them right.
The mistakes that cost marks
- Thinking long arms make a big angle. the size of an angle is how far it turns. The length of the arms does not change it.
- Mixing up acute and obtuse. acute is smaller than a right angle; obtuse is between a right angle and a straight line.
- Naming the wrong angle. Calling a reflex angle “acute” because the arms look close together. follow the curve. If it goes the long way round, the angle is reflex: more than 180°.
- Using 180 for a point. Taking the angles around a point away from 180. a straight line is 180°; all the way round a point is 360°.
- Leaving an angle out. Taking only one of the known angles away. add up every angle you know first, then take the total away.
- Thinking “vertically” means up and down. vertically opposite angles are across the point where two lines cross, whichever way they face.
- Using a fact where it does not fit. Adding angles to 180° when they are not on one straight line. check that the angles meet at one point, on one straight line, before you use 180°.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 7.1.
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