Cambridge Primary Checkpoint · Mathematics · Stage 6 · Geometry and Measure

Angles: Measuring, Drawing and Angles in a Triangle

Classify, estimate, measure and draw angles with a protractor, and use the fact that the angles in a triangle add up to 180°.

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In the curriculum

Classify angles as acute, right, obtuse, straight or reflex; estimate the size of an angle; measure angles with a protractor; and draw angles of a given size. [6Gg.09]
Know that the three angles in any triangle add up to 180°, and use this to calculate a missing angle in a triangle. [6Gg.10]

An angle measures an amount of turn between two lines (arms) that meet at a point (the vertex). We measure angles in degrees (°). A full turn is 360°, half a turn is 180° and a quarter turn is 90°. On this page you will learn to name, estimate, measure and draw angles, and then use the most important angle fact of Stage 6: the angles in a triangle add up to 180°.

1. Key words

Word / phrase	Meaning / use
angle	زاوية — the amount of turn between two arms
vertex	رأس الزاوية — the point where the two arms meet
degree (°)	درجة — the unit for measuring angles
acute angle	زاوية حادة — less than 90°
right angle	زاوية قائمة — exactly 90°
obtuse angle	زاوية منفرجة — more than 90° but less than 180°
straight angle	زاوية مستقيمة — exactly 180° (a straight line)
reflex angle	زاوية منعكسة — more than 180° but less than 360°
protractor	المنقلة — the instrument for measuring and drawing angles
estimate	يقدّر — make a sensible guess using known angles such as 90° and 180°
isosceles triangle	مثلث متساوي الساقين — 2 equal sides and 2 equal angles
equilateral triangle	مثلث متساوي الأضلاع — 3 equal sides and 3 equal angles (each 60°)

2. Classifying angles

The six angle names. The shaded part shows the turn being measured.

Angle	Size	Example
acute	0° < angle < 90°	35°, 72°, 89°
right	= 90°	the corner of a book
obtuse	90° < angle < 180°	91°, 120°, 175°
straight	= 180°	a straight line
reflex	180° < angle < 360°	190°, 270°, 300°
full turn	= 360°	one whole rotation

Estimating. Compare the angle with a right angle (90°) and a straight line (180°). Half a right angle is 45°; a right angle split into three gives 30° and 60°. For example, an angle a little bigger than a right angle might be about 100°; an angle about halfway between a right angle and a straight line is about 135°. Always estimate before you measure — it stops you reading the wrong scale.

3. Measuring an angle with a protractor

The baseline lies along 0° on the inner scale (on the right), so read the inner scale: the angle is 50°, not 130°.

Estimate first: is the angle acute or obtuse?
Put the centre point (cross) of the protractor exactly on the vertex.
Turn the protractor so that the 0° line lies exactly along one arm (the baseline).
Find the scale that starts at 0 on that arm. Follow THAT scale round to the other arm.
Read the number where the second arm crosses the scale. If the arm is short, extend it with a ruler first.
Check against your estimate: an acute angle cannot read 130°.

A protractor has two scales because the angle can open to the right or to the left. In the diagram above, the arm on the right sits on 0 of the inner (blue) scale, so the inner scale gives the answer: 50°. The outer scale shows 130° at the same place — that is 180° − 50°, the angle on the other side.

4. Measuring a reflex angle

A semicircular protractor only goes up to 180°, so measure the smaller angle and subtract from 360°.

The reflex angle and the smaller angle together make a full turn of 360°.

5. Drawing an angle of a given size

To draw an angle of 70° at point A:

Draw a straight line (the first arm) with a ruler, and mark the vertex A at one end.
Place the protractor's centre on A with the 0° line along your arm.
Using the scale that starts at 0 on your arm, count round to 70 and make a small dot.
Remove the protractor and join A to the dot with a ruler.
Mark the angle with an arc and write 70°. Check it looks acute!

To draw a reflex angle such as 250°: work out 360° − 250° = 110°, draw an angle of 110°, then mark the OUTSIDE angle with an arc and label it 250°.

المنقلة بالخطوات

حط مركز المنقلة على رأس الزاوية بالظبط، وخلي خط الصفر على أحد الضلعين.
اقرأ من التدريج اللي بيبدأ بصفر عند الضلع ده — مش من التدريج التاني.
قدّر الأول: لو الزاوية حادة يبقى الإجابة أقل من 90°.

6. The angles in a triangle add up to 180°

Draw any triangle on paper, colour its three corners, tear them off and put them together at one point. They always fit exactly on a straight line — and a straight line is 180°.

The three angles a, b and c of a triangle fit together on a straight line: a + b + c = 180°.

To find a missing angle in a triangle: add the angles you know, then subtract the total from 180°.

Type of triangle	What you know	How to find the missing angles
any triangle	two angles	180° − (sum of the two angles)
right-angled triangle	one angle is 90°	the other two add up to 90°, so missing angle = 90° − known angle
isosceles triangle	the two base angles are equal	if you know the top angle: (180° − top) ÷ 2; if you know a base angle: 180° − 2 × base
equilateral triangle	all three angles are equal	each angle is 180° ÷ 3 = 60°

Two facts that follow: a triangle can have at most one right angle or obtuse angle (two of them would already make 180° or more), and every triangle has at least two acute angles.

أهم قاعدة

مجموع زوايا أي مثلث = 180°. اجمع الزاويتين المعروفتين واطرحهم من 180°.
المثلث متساوي الساقين: زاويتين القاعدة متساويتين. المثلث متساوي الأضلاع: كل زاوية 60°.

7. Worked examples

Worked example 1 (1 mark) Write each angle in the correct column: 118°, 90°, 44°, 270°, 180°, 89°.

acute	right	obtuse	straight	reflex
44°, 89°	90°	118°	180°	270°

Common mistake: putting 89° with the right angles. 89° is less than 90°, so it is acute — even though it looks almost square.

Worked example 2 (1 mark) Use the protractor diagram in section 3. What is the size of the angle?

Estimate: the angle is clearly acute, so the answer is less than 90°.
The baseline (right arm) is on 0 of the inner scale, so use the inner scale.
The other arm crosses the inner scale at 50.
Answer: 50°. (130° would be wrong — it is obtuse.)

Worked example 3 (1 mark) The diagram shows a triangle. Work out the size of angle a.

Triangle with angles 65° and 40° (drawn to scale).

Add the known angles: 65° + 40° = 105°.
Subtract from 180°: 180° − 105° = 75°.
Answer: a = 75°. Check: 65 + 40 + 75 = 180 ✓

Worked example 4 (2 marks) An isosceles triangle has one angle of 52°. The other two angles are equal to each other. Find them.

The 52° angle is the different one, so the two equal angles share 180° − 52° = 128°.
Each equal angle = 128° ÷ 2 = 64°.
Answer: 64° and 64°. Check: 52 + 64 + 64 = 180 ✓
Marks: 1 mark for 180 − 52 (or 128 seen), 1 mark for 64°.

Watch the wording: if the question says the 52° angle is one of the two EQUAL angles, then the angles are 52°, 52° and 180° − 104° = 76°. Read carefully which angle is repeated.

Worked example 5 (1 mark) A right-angled triangle has one angle of 27°. What is the third angle?

The right angle is 90°, so the other two angles add up to 180° − 90° = 90°.
Third angle = 90° − 27° = 63°.
Answer: 63°.

Worked example 6 (1 mark) Omar measured the angles of a triangle and got 64°, 71° and 55°. Explain how you know he has made a mistake.

Answer: 64 + 71 + 55 = 190°, but the angles in a triangle must add up to 180°. So at least one measurement is wrong (they are 10° too much altogether).

Common mistakes

Reading the wrong protractor scale — e.g. writing 130° for an acute angle. Always estimate first.
Putting the protractor's edge, not its centre point, on the vertex.
Not lining up the 0° line exactly along the arm.
Forgetting that a reflex angle is the OUTSIDE angle: measure the inside angle and subtract from 360°.
Adding the angles and forgetting to subtract from 180° (giving 105° instead of 75°).
In an isosceles triangle, not checking which angle is repeated.
Using 360° for a triangle (360° is a full turn — and the angle sum of a quadrilateral — not of a triangle).

How Checkpoint asks this

Ring or tick the correct name (acute, obtuse, reflex…) or sort angles into a table.
Estimate: "Which is the best estimate of this angle?" — compare with 90° and 180°.
Measure an angle printed on the paper with your protractor. The angle is drawn accurately, so measure carefully and extend short arms with a ruler.
Draw an angle of a given size from a line already drawn on the paper.
Calculate a missing angle in a triangle. Diagrams are often marked "not drawn to scale" — then you must calculate, not measure.
Explain why something is impossible, e.g. a triangle with two obtuse angles or angles that add to more than 180°.

Practice

Remember

Acute < 90° · right = 90° · obtuse between 90° and 180° · straight = 180° · reflex between 180° and 360°.
Protractor: centre on the vertex, 0° along one arm, read the scale that starts at 0 on that arm. Estimate first!
Reflex angle = 360° − the smaller angle.
Angles in a triangle add up to 180°: missing angle = 180° − (the other two).
Equilateral: 60° each. Isosceles: two equal base angles. Right-angled: the other two angles add up to 90°.

All pages in this course

Start here

Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

Number: place value, decimals, fractions, percentages and ratio

Geometry and measure

Statistics and probability

Practice tests