Overview
What you’ll learn
- Calculate missing angles on lines, around points, and inside triangles and quadrilaterals, quoting a reason.
- Identify corresponding, alternate and co-interior angles in parallel-line diagrams and state which rule applies.
- Find interior and exterior angles of regular polygons, and the number of sides from one angle.
- Use similarity to scale lengths, areas and volumes (Extended for area and volume factors).
- State line and rotational symmetry orders, and read and construct three-figure bearings.
Syllabus checklist (0580)
Statements are paraphrased and grouped by tier. Core is studied by everyone; Extended content is examined on Papers 2 and 4 only.
Core — every candidate
- 4.6Use angle properties: angles at a point (360°), angles on a straight line (180°), vertically opposite angles, angles of triangles and quadrilaterals.
- 4.6Use angle properties of parallel lines — corresponding, alternate and co-interior — giving reasons.
- 4.6Use interior and exterior angles of regular polygons, including the sums 180(n − 2) and 360.
- 4.5Recognise line and rotational symmetry in 2-D shapes.
- 4.3Interpret scale drawings and use three-figure bearings measured clockwise from north.
Extended — Extended candidates only
- 4.4Use similarity: calculate unknown lengths in similar shapes, and — at Extended — areas (× length²) and volumes (× length³).
In 30 seconds
- 1The bank of facts: straight line 180°, full turn 360°, vertically opposite equal, triangle 180°, quadrilateral 360° — every chase starts here.
- 2Parallel lines: F-angles (corresponding) equal, Z-angles (alternate) equal, C-angles (co-interior) sum to 180° — name the rule in your reason.
- 3Exterior angles of any polygon sum to 360°; a regular polygon’s exterior angle is 360° ⁄ n.
- 4Interior angles of an n-gon sum to 180(n − 2); for a regular polygon, interior = 180° − exterior.
- 5Similar shapes (Extended): lengths × k → areas × k² → volumes × k³.
- 6A shape with rotational symmetry of order n looks the same after each 360° ⁄ n turn; regular polygons have order n.
- 7Bearings are measured clockwise from north, written as three figures: east 090°, south 180°, west 270°.
Notes · 1 of 4
The core angle facts
Five numbers run the whole topic: 90, 180, 360 — and equal pairs.
Angles on a straight line add to 180°: if two angles split a line and one is 118°, the other is 62°. Around a point, they add to 360°: 140 + 90 + missing = 360 gives 130°. When two straight lines cross, the opposite (vertically opposite) angles are equal — the X shape has two pairs.
Triangle angles sum to 180°; quadrilateral angles sum to 360°. An isosceles triangle has two equal sides and two equal base angles: an apex of 40° leaves (180 − 40) ⁄ 2 = 70° for each base angle. The exterior angle of a triangle equals the sum of the two remote interior angles — a fact that turns three-step chases into one-step answers. In every “find the angle” question, attach a reason to the number: the mark scheme pays for the words.
Notes · 2 of 4
Parallel-line angle rules
F, Z and C shapes — with the reason named.
When a line crosses two parallel lines, three families of angles appear. Corresponding angles (the F shape) are equal: one angle of 64° fixes its corresponding partner at 64°. Alternate angles (the Z shape) are equal too. Co-interior angles (the C or U shape, on the same side of the crossing line) sum to 180°: a partner of 72° co-interior makes 108°.
The reason line matters: “corresponding angles are equal” or “co-interior angles sum to 180°” scores the reasoning mark; “they look equal” scores nothing. Mixing the families is the classic error — alternate and corresponding are equal, but co-interior partners are supplementary, not equal.
Notes · 3 of 4
Polygons: interior and exterior angles
Two totals — 360° outside, 180(n − 2) inside.
Walk around any polygon and you turn through one full revolution: the exterior angles sum to 360°, no matter the shape. For a regular polygon all exterior angles match, so each is 360° ⁄ n: a regular hexagon turns 60° per corner. The interior angles of an n-gon sum to 180(n − 2) — a pentagon holds 540°, a decagon 1440°. For regular shapes, interior = 180° − exterior: the pentagon’s interiors are 108° each.
Questions run the machinery both ways: given the exterior angle 24°, the polygon has 360 ⁄ 24 = 15 sides; given a regular polygon with interior 160°, its exterior is 20° and it has 18 sides. The exterior route is almost always faster than dividing 180(n − 2) by n and solving.
Notes · 4 of 4
Similarity, symmetry and bearings
Scale lengths, square for area, cube for volume — and navigate clockwise from north. (Extended for area/volume)
Similar shapes share equal angles and proportional sides. The scale factor is big side ÷ matching small side: if the longest sides are 6 cm and 18 cm, k = 3, and every length of the big shape is 3 times its partner (3, 5, 6 become 9, 15, 18). Extended then squares and cubes the factor: areas × k² (9 times) and volumes × k³ (27 times). A small shape of 6 cm² becomes 54 cm²; a cylinder scaled by 3 holds 27 times as much.
Symmetry comes in two flavours: line symmetry (mirror folds — a rectangle has 2, an equilateral triangle 3) and rotational symmetry (how many times a 360° spin looks identical — order n for a regular n-gon). Bearings are measured clockwise from north as three figures: east is 090°, south 180°, west 270°, south-west 225°. The bearing *of B from A* and *of A from B* differ by exactly 180° — a free check.
Revise
Key terms
The words examiners expect you to use precisely.
- Vertically opposite angles
- Equal angles formed opposite each other where two lines cross.One angle 118° → opposite also 118°.
- Corresponding angles
- Equal F-shape angles on parallel lines.One 64° → corresponding 64°.
- Co-interior angles
- C-shape angles on parallel lines summing to 180°.Partner of 72° is 108°.
- Exterior angle
- The angle between one side and the extension of the next.Exterior angles of any polygon sum to 360°.
- Regular polygon
- A polygon with equal sides and equal angles.Regular hexagon: exterior 60°, interior 120°.
- Similar shapes
- Shapes with equal angles and proportional sides.Scale factor 3 triples every length.
Show 3 more key termsShow fewer key terms
- Area scale factor (Extended)
- The square of the length scale factor.Lengths ×3 → areas ×9.
- Rotational symmetry
- The number of positions in a full turn where a shape looks unchanged.Regular octagon: order 8.
- Three-figure bearing
- Direction measured clockwise from north, written 000°–360°.West is 270°.
Revise
Common mistakes
Where students most often lose marks in this topic — and what to do instead.
Mistake: Treating co-interior angles as equal.
They are supplementary, not equal.
Instead: The co-interior partner of 72° is 108°, since 72 + 108 = 180.
Mistake: Giving an answer with no reason.
The reasoning mark is separate from the value.
Instead: Write “alternate angles are equal” or “angles in a triangle sum to 180°”.
Show 10 more mistakesShow fewer mistakes
Mistake: Using 180(n − 2) for exterior angles.
Exterior angles always total 360°, whatever n is.
Instead: Regular polygon: exterior = 360° ⁄ n.
Mistake: Halving the exterior angle to get the interior.
Interior and exterior sit on a straight line, summing to 180°.
Instead: Regular pentagon: exterior 72° → interior 108°.
Mistake: Cubing the scale factor for areas.
Area grows as length squared.
Instead: Lengths ×3 → areas ×9 → volumes ×27.
Mistake: Measuring bearings anticlockwise or from south.
Bearings run clockwise from north, always three figures.
Instead: West is 270°, not 090° or 180°.
Mistake: Assuming an isosceles apex angle is one of the equal pair.
The apex is the odd one out between the equal sides.
Instead: Apex 40° → base angles (180 − 40)⁄2 = 70° each.
Mistake: Summing interior angles of a quadrilateral as 180°.
Quadrilaterals are two triangles: 360°.
Instead: 130 + 80 + 75 + 75 = 360.
Mistake: Using equal corresponding or alternate angles when the lines are not parallel.
The equalities hold only between parallel lines.
Instead: Check for the parallel arrows on the diagram before quoting any transversal reason.
Mistake: Finding a back bearing by subtracting from 360° every time.
A bearing of 065° would give 295°, which is wrong.
Instead: Add or subtract 180°, whichever lands between 000° and 360°: 065° + 180° = 245°.
Mistake: Reporting 180(n − 2) as the size of each interior angle.
That formula gives the angle sum, not the individual angle.
Instead: Divide by n: a regular octagon has 1080° ÷ 8 = 135° at each corner.
Mistake: Scaling an area by the length scale factor.
Both dimensions grow, so the area grows twice as fast as the lengths.
Instead: Lengths scale by k, areas by k², volumes by k³.
Check yourself
Quick quiz
24 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Find the third angle on a straight line
Two angles on a straight line measure 37° and 90°. The third is
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Practise
Topical questions
14 exam-style questions with model answers and mark schemes.
A set of structured and calculation questions matched to this topic, in three tiers, each with a model answer, mark scheme and examiner tip.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a Cambridge paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Topic 12 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Topic 12: angle facts on lines and around points, triangle and quadrilateral sums, parallel-line rules with reasons, interior and exterior angles of polygons, similarity scale factors, symmetry, and three-figure bearings.
Mark scheme
PROTopic 12 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Which papers examine this topic?
Paper 1 / 3
Core: 80 marks each, 1 h 30 each
Paper 1 non-calculator, Paper 3 calculator
Paper 2 / 4
Extended: 100 marks each, 2 h each
Paper 2 non-calculator, Paper 4 calculator
Accuracy
3 s.f., angles to 1 d.p.
Unless a question states otherwise
Want a full paper? Try the Paper 1-style Core, Paper 2-style non-calculator or Paper 4-style calculator sets.
Questions
Frequently asked questions
Which parallel-line angles are equal and which add to 180°?
Corresponding (F) and alternate (Z) angles are equal. Co-interior (C) angles — the pair on the same side of the transversal between the lines — add to 180°. If you sketch the letter shape each time, the rule follows automatically.
How do I find the number of sides of a regular polygon?
Use the exterior angle: 360° ÷ exterior = number of sides. Given the interior angle instead, subtract from 180° first, then divide 360° by the result. This route is much faster than solving 180(n − 2) ⁄ n = interior.
Why do areas scale by the square and volumes by the cube?
Area is two lengths multiplied, so the factor appears twice: k × k = k². Volume is three lengths multiplied, so k³. For a scale factor of 3, lengths triple, areas go up nine-fold, and volumes twenty-seven-fold.
How do I write a bearing correctly?
Always measure clockwise from north at the point named after “from”, and always write three figures: 045°, not 45°. The bearing of A from B and of B from A differ by 180° — handy as a sanity check.
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