Measuring and Drawing Lines and Angles

Use a ruler, protractor and set square to measure and draw lines and angles accurately, find a reflex angle, and read the number at an arrow on a marked scale.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1 · 18 real past-paper questions below

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Three Tools for Accuracy

Some exam marks are for accuracy alone: measure this, draw that. They are quick marks if you use the right tool.

Tool	Use it for
Ruler	lengths, to the nearest mm
Protractor	angles, to the nearest degree
Set square	right angles and parallel lines

Rule

Mark schemes usually accept 2 mm or $2^{\circ}$ either way. Aim to be exact, so a small slip still scores.

Measure a Line

The 1-mark question: measure the line $A B$ in centimetres or millimetres. It is the most common measuring question.

Worked example

1
$0 of the ruler on A$ — line up the zero mark, not the end of the plastic
2
$at B : 7 cm and 4 mm$ — read the last whole centimetre, then count the small marks
3
$A B = 7.4 cm$ — 4 mm is 0.4 cm
4
$A B = 74 mm$ — times 10 when the question wants millimetres

Common mistake

Starting at the end of the ruler, or at 1. Every length then comes out wrong by the same amount.

Key idea

Write the unit the question asks for. If it says "give the units", a bare 7.4 loses a mark.

Which Protractor Scale?

The 1-mark question: measure angle $x$ . A protractor has two scales running opposite ways, so the second arm points at two numbers.

Steps

1
Put the centre cross exactly on the corner of the angle.
2
Line up the zero line with one arm.
3
Follow the scale that starts at 0 on that arm, round to the other arm.

Worked example

1
$readings: 5 7^{\circ} and 12 3^{\circ}$ — both numbers at the second arm
2
$5 7^{\circ} + 12 3^{\circ} = 18 0^{\circ}$ — the two scales always add to 180°
3
$wider than 9 0^{\circ}$ — look before you read: it is obtuse
4
$x = 12 3^{\circ}$ — the obtuse reading

Common mistake

Reading the other scale and writing 57°. One look tells you acute or obtuse, and that picks the number.

Measuring a Reflex Angle

The 1-mark question: measure an angle marked round the outside of a corner. A protractor only reaches $18 0^{\circ}$ , so measure the small angle instead.

Worked example

1
$small angle = 3 4^{\circ}$ — measure the angle inside the corner
2
$36 0^{\circ} - 3 4^{\circ}$ — the two angles make a full turn
3
$= 32 6^{\circ}$ — the reflex angle

Common mistake

Writing 34° as the answer. An arc drawn round the outside means reflex: more than 180°.

Draw an Angle or a Line

The 2-mark question: draw an angle of a given size, or a line of a given length. Often one line is drawn for you.

Steps

1
A line: put the 0 on your start point, mark a dot at the length, then rule from the start to the dot.
2
An angle: put the centre cross on the point and the zero line along the given line.
3
Count up from the 0 on that line to the size you need, and mark a dot.
4
Take the protractor away and rule from the point through the dot.

Worked example

1
$angle A B C = 6 5^{\circ} at B$ — the task, with the line AB drawn
2
$count from 0 on B A$ — use the scale that is 0 along the given line
3
$dot at 6 5^{\circ}$ — it must look narrower than a square corner
4
$measure it back: 6 5^{\circ}$ — a quick check before you move on

Common mistake

Marking the dot at 115° by counting on the wrong scale. A 65° angle must look acute.

Parallel and Perpendicular with a Set Square

The 1-mark question: draw a line through $P$ that is perpendicular (at a right angle) or parallel to a given line. You need a ruler and a set square, not compasses.

Steps

1
Perpendicular: put one short edge of the set square along the line. Slide it until the other short edge passes through P, then rule along it.
2
Parallel: put one edge of the set square on the line and hold a ruler firmly against another edge.
3
Slide the set square along the ruler until that edge reaches P, then rule along it.

Key idea

Check a parallel line by measuring the gap at both ends: it should be the same. Check a perpendicular with the protractor: it should read 90°.

Common mistake

Drawing the line freehand, or by eye. The mark needs a ruled line at the right angle.

Reading a Marked Scale

The 1-mark question: write down the number at the arrow on a marked scale. Work out what one small gap is worth first.

Worked example

1
$3 - 2 = 1 kg$ — the difference between two labels
2
$1 \div 5 = 0.2 kg$ — 5 gaps between the labels: count gaps, not marks
3
$2 + 3 \times 0.2$ — the arrow is 3 gaps past the 2
4
$= 2.6 kg$ — the reading

Common mistake

Taking every small gap as 1, or as 0.1, without checking. Always work it out from the labels.

Which Method? Read the Question

Read the command word first: measure, draw, or write down.

You see	You do
Measure the line in mm	ruler from 0, then times 10
Measure the angle	look first, then pick the scale
Arc round the outside	360° take away the small angle
Draw an angle of a size	count from 0 on the given line
Draw a parallel or perpendicular	set square and ruler
Perpendicular on a grid	Perpendicular and Parallel Lines
Name the type of angle	Types of Angle
Ruler and compasses only	Constructing Triangles
Measure, then use the scale	Scale Drawings
Measure a bearing	Three-Figure Bearings

Key idea

Use a sharp pencil. The examiner checks your drawing with a ruler and protractor, and a thick line can be 1 mm out by itself.

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