Angles
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Revision notes for CIE IGCSE Maths Angles. Open the guide for explanations and worked examples. Written against the CIE IGCSE Maths (0580) specification, so the content matches what's examinable rather than general Maths background.

Angles

What you'll learn

  • How to recognise acute, right, obtuse and reflex angles.
  • How to measure and draw angles using a protractor.
  • How to find missing angles on straight lines, around points and in triangles.
  • How to use isosceles triangles, equilateral triangles and squares in angle questions.

What is an angle?

An angle measures a turn between two straight lines that meet at a point. The point where they meet is called the vertex.

Angles are measured in degrees, written with the symbol °.

Definition

Key angle types

  • An acute angle is less than 90°.
  • A right angle is exactly 90°.
  • An obtuse angle is more than 90° but less than 180°.
  • A reflex angle is more than 180° but less than 360°.

This diagram shows the four common angle types you need to recognise quickly.

Four common angle types: acute, right, obtuse and reflex

Example

Naming angles from a diagram

A diagram shows four angles labelled P, Q, R and S. P is 90°, Q is a small angle, R is wider than 90° but not a straight line, and S is the large angle going the long way round. Match each label to its angle type.

  1. P is exactly 90°, so P is a right angle.

  2. Q is smaller than 90°, so Q is an acute angle.

  3. R is bigger than 90° but smaller than 180°, so R is an obtuse angle.

  4. S is bigger than 180°, so S is a reflex angle.

Tip

Quick check

Compare the angle to a corner of a rectangle. Smaller than a corner means acute; exactly a corner means right; bigger than a corner but not straight means obtuse.

Measuring angles with a protractor

A protractor is the semicircle tool marked from 0° to 180°.

To measure an angle:

  1. Put the centre of the protractor exactly on the vertex.
  2. Line up one arm of the angle with the 0° line.
  3. Read the scale where the other arm crosses the protractor.
  4. Choose the scale that starts from the arm you lined up with 0°.
Common Mistake

Reading the wrong scale

Most protractors have two scales. If your angle is clearly acute but you read something like 140°, you have probably used the wrong scale.

Example

Measuring an obtuse angle

An angle opens wider than a right angle. You place a protractor on it and the second arm reaches the mark labelled 116° on the correct scale. Classify the angle.

  1. Check that the protractor centre is on the vertex.

  2. Check that one arm is lined up with 0°.

  3. Read the correct scale: the angle is 116°.

  4. Since 116° is more than 90° but less than 180°, the angle is obtuse.

Drawing angles

To draw an angle, you use the protractor in reverse.

Example

Drawing an angle of 110°

Draw an angle of 110° and label it B.

  1. Draw a straight line segment. This will be one arm of the angle.

  2. Put a point at one end of the line segment. This point is the vertex.

  3. Place the centre of the protractor on the vertex and line up the 0° line with your first arm.

  4. Find 110° on the correct scale and make a small mark.

  5. Remove the protractor and draw a straight line from the vertex through your mark.

  6. Label the angle B.

Tip

Drawing obtuse angles

110° is bigger than 90°, so your angle should look wider than a right angle. If it looks tiny, you used the wrong scale.

Angles on a straight line

A straight line angle is 180°. If angles sit next to each other on a straight line, they add to 180°.

Key Idea

Straight line fact

Angles on a straight line add up to 180°.

The three facts below are the ones you will use most often in this topic.

Angle facts: straight line, around a point, and isosceles triangle

Example

Finding an angle on a straight line

A ray splits a straight line into two angles. One angle is 128°. The other angle is marked xxx. Find xxx.

  1. Angles on a straight line add to 180°.

  2. Write the equation:

    x+128=180x + 128 = 180x+128=180

  3. Subtract 128 from 180:

    x=180−128=52x = 180 - 128 = 52x=180−128=52

  4. The missing angle is 52°.

Example

Giving a reason

Two adjacent angles on a straight line are 137° and xxx. Find xxx and give a reason.

  1. Use the straight line fact:

    x+137=180x + 137 = 180x+137=180

  2. Subtract:

    x=180−137=43x = 180 - 137 = 43x=180−137=43

  3. The answer is 43°. The reason is: angles on a straight line add up to 180°.

Right angles and perpendicular lines

A right angle is 90°. Two lines are perpendicular if they meet at a right angle.

Definition

Perpendicular lines

Perpendicular lines meet at 90°.

Sometimes a right angle is split into two smaller angles. Those two angles must add to 90°.

Example

A right angle split into two parts

Two perpendicular lines form a right angle. A ray inside the right angle makes an angle of 34° with one line. The other angle is marked xxx. Find xxx.

  1. Perpendicular lines make 90°.

  2. The two smaller angles add to 90°:

    x+34=90x + 34 = 90x+34=90

  3. Subtract 34 from 90:

    x=90−34=56x = 90 - 34 = 56x=90−34=56

  4. The missing angle is 56°.

Angles around a point

A full turn is 360°. So all the angles around one point add to 360°.

Key Idea

Around a point

Angles around a point add up to 360°.

Example

Finding a missing angle around a point

Several rays meet at one point. Three of the angles are 90°, 105° and 118°. The remaining angle is marked xxx. Find xxx.

  1. Angles around a point add to 360°.

  2. Add the known angles:

    90+105+118=31390 + 105 + 118 = 31390+105+118=313

  3. Subtract from 360:

    x=360−313=47x = 360 - 313 = 47x=360−313=47

  4. The missing angle is 47°.

Common Mistake

Using 180° instead of 360°

Use 180° only when the angles lie on a straight line. If the angles go all the way around a point, use 360°.

Angles in triangles

A triangle is a three-sided shape. The three inside angles of any triangle always add to 180°.

Key Idea

Triangle angle sum

Angles in a triangle add up to 180°.

Example

Finding the third angle in a triangle

In triangle ABC, angle A is 58° and angle C is 41°. Find angle B.

  1. Angles in a triangle add to 180°.

  2. Add the two known angles:

    58+41=9958 + 41 = 9958+41=99

  3. Subtract from 180:

    ∠B=180−99=81\angle B = 180 - 99 = 81∠B=180−99=81

  4. Angle B is 81°.

Tip

Reason wording

If a question asks for a reason, write: angles in a triangle add up to 180°. That is clear and exam-friendly.

Isosceles and equilateral triangles

An isosceles triangle has two equal sides. The angles opposite those equal sides are equal too. These equal angles are often called the base angles.

An equilateral triangle has three equal sides. All three angles are 60°.

Definition

Isosceles triangle fact

In an isosceles triangle, equal sides have equal opposite angles.

Example

Using an isosceles triangle

An isosceles triangle has two equal sides meeting at the top. One base angle is 72°. The top angle is marked xxx. Find xxx.

  1. In an isosceles triangle, the base angles are equal.

  2. So the other base angle is also 72°.

  3. Angles in a triangle add to 180°:

    x+72+72=180x + 72 + 72 = 180x+72+72=180

  4. Subtract the two base angles from 180:

    x=180−144=36x = 180 - 144 = 36x=180−144=36

  5. The top angle is 36°.

Common Mistake

Equal sides do not mean every angle is equal

Only an equilateral triangle has all three angles equal. An isosceles triangle usually has just two equal angles.

Squares, equilateral triangles and combined diagrams

In combined shape questions, you often need to know the angle facts for common shapes.

  • In a square, every angle is 90°.
  • In an equilateral triangle, every angle is 60°.
  • If two shapes share a side, look carefully at the angles meeting at the same point.
Example

Square joined to an equilateral triangle

A square ABDE has an equilateral triangle BCD attached to side BD. Find angle ADC.

  1. Angle ADB is a corner of the square, so it is 90°.

  2. Angle BDC is an angle in an equilateral triangle, so it is 60°.

  3. Angle ADC is made from angle ADB and angle BDC together:

    ∠ADC=90+60=150\angle ADC = 90 + 60 = 150∠ADC=90+60=150

  4. Angle ADC is 150°.

Multi-step angle problems

Some questions combine two or three facts. The best method is to find one angle at a time and write a reason for each step.

Example

Using a straight line and an isosceles triangle

Points A, B and C lie on a straight line. D is above the line. Triangle ABD is isosceles with AD = BD. Angle ADB is 64°. Angle DCB is 31°. Find angle BDC.

  1. In triangle ABD, AD = BD, so the base angles at A and B are equal.

  2. Find the two equal base angles. First subtract the top angle from 180:

    180−64=116180 - 64 = 116180−64=116

  3. Split 116 equally between the two base angles:

    116÷2=58116 \div 2 = 58116÷2=58

  4. Since A, B and C lie on a straight line, angle DBC and angle ABD add to 180°:

    ∠DBC=180−58=122\angle DBC = 180 - 58 = 122∠DBC=180−58=122

  5. Now use triangle DBC. The angles in a triangle add to 180°:

    ∠BDC=180−122−31=27\angle BDC = 180 - 122 - 31 = 27∠BDC=180−122−31=27

  6. Angle BDC is 27°.

Showing a triangle is isosceles

Sometimes you are asked to show that a triangle is isosceles. This means you must prove that two angles are equal, or two sides are equal.

At this level, you will usually prove two angles are equal. If two angles in a triangle are equal, then the sides opposite them are equal, so the triangle is isosceles.

Example

Showing a triangle is isosceles

Triangle ABD sits on a straight line ABC. Angle ADB is 70°. The exterior angle CBD is 125°. Show that triangle ABD is isosceles.

  1. Since ABC is a straight line, angles ABD and CBD add to 180°.

  2. Find angle ABD:

    ∠ABD=180−125=55\angle ABD = 180 - 125 = 55∠ABD=180−125=55

  3. Use the angles in triangle ABD to find angle DAB:

    ∠DAB=180−70−55=55\angle DAB = 180 - 70 - 55 = 55∠DAB=180−70−55=55

  4. Angle DAB and angle ABD are both 55°.

  5. Therefore triangle ABD has two equal angles, so it is isosceles.

Tip

Diagrams may not be accurate

Do not guess from how the diagram looks. Use the angle facts and calculate carefully.

Exam technique

In the exam

  1. Mark known angle facts on the diagram first: straight line 180°, right angle 90°, full turn 360°, triangle 180°.

  2. If asked to give a reason, write the exact fact you used, such as “angles on a straight line add up to 180°”.

  3. For multi-step questions, find one missing angle at a time and show your working clearly.

Self review

Check yourself

  • Can you explain the difference between acute, obtuse and reflex angles?
  • If two angles lie on a straight line and one is 143°, what calculation finds the other?
  • In an isosceles triangle, which angles are equal and why?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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