Mathematics 0580 · syllabus C4.1, C4.6, E4.1, E4.6
Cambridge IGCSE Mathematics 0580 — Angles, parallel lines and polygons topical past papers
Angle facts at points and on lines, angles in parallel lines, triangles and quadrilaterals, interior and exterior angles of polygons, and geometrical terms.
- Questions
- 384
- Years
- 2009–2026
- With mark scheme
- 93%
- Questions 2022–2026
- 177
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Angles, parallel lines and polygons covers
This topic is the foundation of geometry. You need the vocabulary: acute, obtuse and reflex angles, parallel and perpendicular lines, the types of triangle, the special quadrilaterals (square, rectangle, rhombus, parallelogram, kite and trapezium) and their properties, polygons and their names, and the faces, edges and vertices of simple solids. Then come the angle facts: angles at a point add up to 360°, angles on a straight line to 180°, vertically opposite angles are equal, and the angles of a triangle add up to 180° and those of a quadrilateral to 360°.
Where a line crosses two parallel lines, corresponding angles are equal, alternate angles are equal and co-interior angles add up to 180°. For polygons, the exterior angles add up to 360°, so each exterior angle of a regular polygon with n sides is 360° ÷ n, and the interior angles add up to (n − 2) × 180°. Core works with regular polygons; Extended includes irregular ones, such as a pentagon with some angles given in terms of x. Angles are named with three letters, so angle ABC is the angle at B.
Questions ask you to give reasons, and a reason must use the correct geometrical words, such as 'alternate angles' or 'angles on a straight line'. This is Core content, so it can be set on all four papers, and its facts are used throughout geometry: in circle theorems, bearings, similar triangles and trigonometry, a missing angle is often found with these facts first.
In the syllabus
The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.
- C4.1 Geometrical terms
- C4.6 Angles
- E4.1 Geometrical terms
- E4.6 Angles
What you should be able to do
- Use the correct terms for angles, lines, triangles, quadrilaterals, polygons and solids
- Calculate unknown angles at a point, on a straight line, and in triangles and quadrilaterals
- Find angles formed with parallel lines and name the property used
- Find interior and exterior angles of regular polygons, and the number of sides from an angle
- Use the angle sum of an irregular polygon to form and solve an equation (Extended)
- Give a geometrical reason for each step of an angle calculation
Key ideas
- Angles on a straight line add up to 180°, angles at a point to 360°, and vertically opposite angles are equal
- With parallel lines, corresponding and alternate angles are equal and co-interior angles add up to 180°
- The interior angles of an n-sided polygon add up to (n − 2) × 180°
- The exterior angles of a polygon add up to 360°; for a regular polygon each one is 360° ÷ n
- At each vertex, interior angle + exterior angle = 180°
How Angles, parallel lines and polygons has been examined
From 2022 to 2026, Angles, parallel lines and polygons was on Paper 3 · Calculator (Core) in 13 of 14 exam series, worth about 11% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Non-calculator (Core) | 57 | 7% | 12 of 14 |
| Paper 2 · Non-calculator (Extended) | 46 | 5% | 13 of 14 |
| Paper 3 · Calculator (Core) | 47 | 11% | 13 of 14 |
| Paper 4 · Calculator (Extended) | 27 | 5% | 9 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 3.
Filed in Momentum under: Angles, Parallel Lines, Polygons.
A worked example
Each interior angle of a regular polygon is 156°. Find the number of sides. An irregular pentagon has angles 2x, 3x − 10°, 100°, 110° and x + 40°. Find x.
- Each exterior angle of the regular polygon is 180° − 156° = 24°.
- The exterior angles add up to 360°, so the number of sides is 360 ÷ 24 = 15.
- A pentagon has 5 sides, so its interior angles add up to (5 − 2) × 180° = 540°.
- 2x + (3x − 10) + 100 + 110 + (x + 40) = 540, so 6x + 240 = 540 and x = 50.
- Check: the angles are 100°, 140°, 100°, 110° and 90°, which add up to 540°.
Answer: 15 sides; x = 50
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Giving 'Z angles' or 'F angles' as reasons: mark schemes want the terms alternate angles and corresponding angles.
- Taking 360° ÷ n as the interior angle: it is the exterior angle, so each interior angle of a regular octagon is 180° − 45° = 135°, not 45°.
- Assuming lines are parallel, or that a triangle is isosceles, because the diagram looks that way: diagrams are not drawn to scale, so use only what is given or proved.
- Measuring an angle on a diagram marked 'not to scale': the answer has to be calculated.
- Giving the answer without reasons when the question asks for them: the reasons carry marks of their own.
Exam technique
- Mark each angle you find on the diagram as you go, and write its reason beside each step.
- For regular polygons, start with the exterior angle, 360° ÷ n; it is quicker and safer than the interior angle sum.
- Look for straight lines and parallel lines first; most unknown angles follow from them in one or two steps.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.