Types of Angle

Name angles as acute, right, obtuse, straight or reflex, measure them with a protractor, and use 90°, 180° and 360° with the right reason.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 44 real past-paper questions below

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Five Names for Angles

An angle measures a turn between two lines that meet at a corner. It is measured in degrees. A full turn is $36 0^{\circ}$ .

Name	Size
Acute	less than $9 0^{\circ}$
Right	exactly $9 0^{\circ}$ (a square corner)
Obtuse	between $9 0^{\circ}$ and $18 0^{\circ}$
Straight	exactly $18 0^{\circ}$ (a straight line)
Reflex	between $18 0^{\circ}$ and $36 0^{\circ}$

Memory trick

Acute is a cute little angle. Obtuse is wider than a square corner. Reflex goes round the back.

Key idea

Why is a full turn 360°? Many historians trace it to the ancient Babylonians, in what is now Iraq, who counted in 60s. 360 is also close to the number of days in a year, and it splits evenly into halves, thirds, quarters, fifths, sixths, eighths, tenths and twelfths.

Name or Mark the Angle

The 1-mark question: write the name of an angle, or mark one on a shape with a letter.

Worked example

1
$6 4^{\circ} < 9 0^{\circ}$ — 64° is less than 90°, so it is smaller than a square corner. That makes it acute.
2
$9 0^{\circ} < 12 8^{\circ} < 18 0^{\circ}$ — 128° is more than a square corner but less than a straight line. That makes it obtuse.
3
$25 0^{\circ} > 18 0^{\circ}$ — 250° turns further than a straight line. That makes it reflex.

To mark an angle with a letter, for example “mark a right angle with R”:

Steps

1
Draw a small arc across the corner, between the two lines.
2
Write the letter next to the arc, so it is clear which corner you mean.
3
For a right angle you can draw a small square in the corner instead.

Key idea

Hold the corner of a sheet of paper against the angle. Inside the corner is acute. Sticking out past it is obtuse.

Common mistake

Marking the small angle when asked for a reflex one. Reflex is the big turn, more than 180°. At a pointed corner of a shape, draw its arc round the outside.

Reading Angle ABC

Exams often name an angle with three letters, like “measure angle ABC”. Here is how to find it fast.

Rule

In angle $A B C$ , the middle letter B is the corner. The angle is between the lines BA and BC. It can also be written $∠ A B C$ .

Worked example

1
$∠ Q R S$ — Look at the middle letter, R. The angle sits at corner R of the shape.
2
$arms: R Q and R S$ — The first and last letters give the two arms: the line from R to Q and the line from R to S.
3
$∠ QR S = ∠ S R Q$ — Reading the letters backwards gives the same angle, because R is still in the middle.
4
$centre cross on R$ — Now measure as usual, with the protractor's centre on R and the zero line along RQ.

Key idea

If only one angle meets at a corner, a single letter is enough: “angle B”.

Common mistake

Measuring at A for angle ABC, because A comes first. The corner is always the middle letter.

Measuring With a Protractor

The 1-mark question: measure the angle marked $x$ . You must be within $2^{\circ}$ .

Steps

1
Put the centre cross of the protractor exactly on the corner.
2
Line up the zero line with one arm of the angle.
3
Count up from the 0 on that arm's scale to the other arm.

Worked example

1
$estimate: acute, so under 9 0^{\circ}$ — Look before you measure. The angle is narrower than a square corner, so the answer must be less than 90°.
2
$the scales show 4 2^{\circ} and 13 8^{\circ}$ — A protractor has two scales, so the second arm passes two numbers. They always add to 180°: 42 + 138 = 180.
3
$x = 4 2^{\circ}$ — Pick the reading that matches your estimate. 42° is acute, while 138° is obtuse.

Common mistake

Reading the wrong scale. Decide acute or obtuse before you read.

Key idea

A reflex angle: measure the small angle, then take it from $36 0^{\circ}$ . A small angle of $14 5^{\circ}$ gives $36 0^{\circ} - 14 5^{\circ} = 21 5^{\circ}$ .

To draw an angle, for example $6 5^{\circ}$ , do the same in reverse:

Steps

1
Draw one arm with a ruler and mark the corner at one end.
2
Put the centre cross on the corner and the zero line along the arm.
3
Count up from that arm's 0 to 65 and make a small dot.
4
Join the corner to the dot with a ruler. Label the angle.

Key idea

If the question only says “draw an acute angle”, any angle clearly less than 90° scores the mark.

Using 90°, 180° and 360°

The 2-mark question: work out an angle, then give a reason. The reason is a sentence the examiner is looking for.

Worked example

1
$x + 5 8^{\circ} = 18 0^{\circ}$ — x and 58° sit side by side on a straight line, and angles on a straight line add up to 180°.
2
$x = 18 0^{\circ} - 5 8^{\circ}$ — Take 58° from both sides to get x on its own.
3
$x = 12 2^{\circ}$ — 180 − 58 = 122. Check: 122 + 58 = 180, so it fits the line.

You see	Reason to write
A square corner	a right angle is 90°
Angles on a line	angles on a straight line add up to 180°
Angles all round a point	angles at a point add up to 360°

Key idea

The reason usually earns its own mark. A correct angle with no reason, or a vague one, loses that mark.

Common mistake

Writing "it is a straight line". Give the whole rule, with the number.

Which Method? Read the Question

Is the question asking for a name, a measurement, or a calculation?

You see	You do
Write down the type	compare with 90° and 180°
Mark a reflex angle	the big turn, over 180°
Angle ABC	the corner is the middle letter, B
Measure the angle	protractor, check acute or obtuse
Work out, give a reason	Angles at a Point and on a Line
Angles in a triangle	Angles in a Triangle
A reason at each stage	Multi-Step Angle Reasoning

Key idea

Always estimate first. If your protractor says 138° but the angle looks acute, you read the wrong scale.

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