Edexcel IGCSE Maths (4MA1) · Unit 5: Geometry

Edexcel IGCSE 4MA1 Angles and Polygons: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

Angles on a straight line add to $180^{\circ}$ , angles round a point to $360^{\circ}$ and the angles in a triangle to $180^{\circ}$ . With parallel lines, alternate and corresponding angles are equal and co-interior angles add to $180^{\circ}$ . In any polygon the exterior angles add to $360^{\circ}$ and the interior angles add to $(n - 2) \times 180^{\circ}$ , so each interior angle of a regular hexagon is $120^{\circ}$ .

Unit 5: Geometry
Higher tier · calculator allowed
Free, no sign-in

Practise angles and polygons in Unit 5 →

Notes
Formulas
Worked examples
Mistakes
FAQ

Angles and Polygons notes

Angle facts

Angles on a straight line add to $180^{\circ}$ , angles round a point add to $360^{\circ}$ and vertically opposite angles are equal. When a question asks for reasons, give the fact in words with every angle you find, such as "angles on a straight line add up to $180^{\circ}$ ".

Angles and parallel lines

Where a line crosses two parallel lines, alternate angles (a Z shape) are equal, corresponding angles (an F shape) are equal and co-interior angles (a C shape) add to $180^{\circ}$ . If one angle is $65^{\circ}$ , its co-interior partner is $115^{\circ}$ .

Triangles

The angles in a triangle add to $180^{\circ}$ . An isosceles triangle has two equal sides and two equal base angles; an equilateral triangle has three angles of $60^{\circ}$ . The exterior angle of a triangle equals the sum of the two interior angles opposite it.

Interior and exterior angles of polygons

The interior angles of an $n$ -sided polygon add to $(n - 2) \times 180^{\circ}$ : $540^{\circ}$ for a pentagon and $1440^{\circ}$ for a decagon. The exterior angles of any polygon add to $360^{\circ}$ , so a regular polygon has exterior angle $\frac{360^{\circ}}{n}$ and interior angle $180^{\circ}$ minus that: a regular hexagon has exterior angle $60^{\circ}$ and interior angle $120^{\circ}$ .

Finding the number of sides

Divide $360^{\circ}$ by the exterior angle. If each interior angle of a regular polygon is $156^{\circ}$ , each exterior angle is $24^{\circ}$ , so the polygon has $\frac{360}{24} = 15$ sides.

Key formulas

Polygons ( $n$ sides) Learn it: $interior sum = (n - 2) \times 180^{\circ}, exterior angle (regular) = \frac{360^{\circ}}{n}$

Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Angles and Polygons worked examples

Worked example 1: Co-interior angles

Two co-interior angles between parallel lines are $(x + 20)^{\circ}$ and $(2 x + 10)^{\circ}$ . Find both angles.

Two parallel lines cut by a transversal; co-interior angles (x + 20) and (2x + 10) degrees (not accurately drawn)

Solution

Co-interior angles add to $180^{\circ}$ : $3 x + 30 = 180$ , $x = 50$ .
Angles: $70^{\circ}$ and $110^{\circ}$ .

3 steps

Tip: Alternate and corresponding angles are equal; co-interior add to 180°.

Worked example 2: Interior angle of a regular polygon

Find each interior angle of a regular octagon.

Solution

Exterior angle $= 360 \div 8 = 45^{\circ}$ .
Interior $= 180 - 45 = 135^{\circ}$ .

3 steps · 1 to fill in

Tip: Exterior angles of any polygon add to 360°.

Worked example 3: Angles in an irregular heptagon · 4 marks

The interior angles of a heptagon are $2 x^{\circ}$ , $2 x^{\circ}$ , $(x + 40)^{\circ}$ , $(x + 40)^{\circ}$ , $150^{\circ}$ , $130^{\circ}$ and $120^{\circ}$ . (a) Find the value of $x$ . (b) Write down the size of the largest angle of the heptagon.

Mark scheme

M1 angle sum (7 − 2) × 180 = 900
M1 6x + 80 + 400 = 900
A1 x = 70
B1 150° (the angles are 140°, 140°, 110°, 110°, 150°, 130°, 120°)

Answer: (a) $x = 70$ (b) 150°

Free account: +5 practice questions on every topic, with instant feedback, and full worked answers on the free questions. Free, no card. Join free →

Angles and Polygons exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

ABC is an isosceles triangle with AB = AC. Angle BAC = 46°. Work out the size of angle ABC.

Show the answer and mark scheme

M1 (180 − 46) ÷ 2
A1 67°

Answer: 67°

Question 2 · medium · 4 marks

AB and CD are parallel lines, with A and C on the left. A straight line crosses AB at P and CD at Q, with Q below and to the right of P. Angle $B P Q = (5 x - 12)^{\circ}$ and angle $P Q C = (3 x + 24)^{\circ}$ . (a) Find the value of $x$ . (b) Work out the size of angle APQ. Give reasons for your answers.

Full worked solution Free, no sign-in

M1 5x − 12 = 3x + 24 (alternate angles are equal)
A1 x = 18
B1 102°
B1 reasons: alternate angles are equal; angles on a straight line add up to 180°

Answer: (a) $x = 18$ (b) 102°

Every question here has a full worked solution. See them all →

Question 3 · medium · 3 marks

In triangle PQR, angle P = $(2 x + 10)^{\circ}$ , angle Q = $(x + 40)^{\circ}$ and angle R = $(3 x - 50)^{\circ}$ . (a) Find the value of $x$ . (b) Show that triangle PQR is isosceles.

Show the answer and mark scheme

M1 2x + 10 + x + 40 + 3x − 50 = 180
A1 x = 30
A1 P = 70°, Q = 70°, R = 40°: two equal angles, so isosceles (shown)

Answer: (a) $x = 30$ (b) angles 70°, 70° and 40°, so isosceles (shown)

Question 4 · medium · 2 marks

Each interior angle of a regular polygon is 162°. Work out the number of sides of the polygon.

Show the answer and mark scheme

M1 exterior angle 180 − 162 = 18, then 360 ÷ 18
A1 20

Answer: 20

Question 5 · hard · 5 marks

PQ and RS are parallel lines, with PQ above RS and P, R on the left. B is a point on PQ and D is a point on RS. C is a point between the two lines, to the right of both B and D. Angle $Q B C = (2 y + 10)^{\circ}$ , angle $S D C = (3 y - 5)^{\circ}$ and angle $B C D = 110^{\circ}$ (the angle at C between CB and CD, not the reflex angle). (a) Find the value of $y$ . (b) Work out the size of angle PBC. Give reasons for your answers.

Show the answer and mark scheme

M1 draws a line through C parallel to PQ, so angle BCD = angle QBC + angle SDC (alternate angles)
M1 2y + 10 + 3y − 5 = 110
A1 y = 21
B1 angle PBC = 180 − 52 = 128°
B1 reasons: alternate angles are equal; angles on a straight line add up to 180°

Answer: (a) $y = 21$ (b) 128°

Question 6 · hard · 4 marks

Each interior angle of regular polygon A is 5 times its exterior angle. Regular polygon B has 6 more sides than polygon A. Work out how much larger an interior angle of B is than an interior angle of A.

Show the answer and mark scheme

M1 e + 5e = 180, so e = 30
A1 A has 360 ÷ 30 = 12 sides
M1 B has 18 sides: interior angle 180 − 360 ÷ 18 = 160
A1 160 − 150 = 10°

Answer: 10°

Question 7 · hard · 5 marks

The interior angles of a polygon add up to $2340^{\circ}$ . (a) Work out the number of sides of the polygon. (b) Eleven of the interior angles are each $150^{\circ}$ . The remaining angles are all equal. Work out the size of each remaining angle. (c) Explain why no polygon can have interior angles that add up to $2000^{\circ}$ .

Show the answer and mark scheme

M1 2340 ÷ 180 + 2 or (n − 2) × 180 = 2340
A1 15
M1 (2340 − 11 × 150) ÷ 4
A1 172.5°
B1 2000 ÷ 180 = 11.1…, not a whole number, so (n − 2) × 180 = 2000 has no whole-number solution

Common angles and polygons mistakes

Mixing up the parallel-line facts. Alternate and corresponding angles are equal; co-interior angles add to 180°, so they are only equal when both are 90°.
Using the interior sum for the exterior angles. The exterior angles of every polygon add to 360°, whatever the number of sides. (n − 2) × 180° is the sum of the interior angles.
Treating an irregular polygon as regular. Only a regular polygon has equal angles. For an irregular polygon, find the angle sum and subtract the angles you know.
Angles without reasons. When a question asks for reasons, name each angle fact in words, such as alternate angles are equal. A correct number on its own does not explain how you got it.

Where students lose marks across 4MA1, from the examiner reports →

Angles and Polygons FAQ

How do I find the interior angle of a regular polygon?

Divide 360° by the number of sides to get the exterior angle, then subtract it from 180°. For a regular decagon, 360 ÷ 10 = 36°, so each interior angle is 144°.

Is the polygon angle formula on the 4MA1 formula sheet?

No. The interior angle sum (n − 2) × 180° and the exterior angle sum of 360° are not printed on the 4MA1 Higher formulae sheet, so learn them.

Can I use a calculator for angles and polygons in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Angles and Polygons revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 angles questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

Practise angles and polygons with your progress saved

Save your answers and track your progressA free account remembers the questions you have done and brings you back to this page.

Join free

Already have an account? Sign in

Edexcel IGCSE 4MA1 Angles and Polygons: exam questions and answers, notes and worked examples

Try it first

Angles and Polygons notes

Angle facts

Angles and parallel lines

Triangles

Interior and exterior angles of polygons

Finding the number of sides

Key formulas

Angles and Polygons worked examples

Worked example 1: Co-interior angles

Solution

Worked example 2: Interior angle of a regular polygon

Solution

Worked example 3: Angles in an irregular heptagon · 4 marks

Mark scheme

Angles and Polygons exam-style questions

Question 1 · easy · 2 marks

Question 2 · medium · 4 marks

Full worked solution Free, no sign-in

Question 3 · medium · 3 marks

Question 4 · medium · 2 marks

Question 5 · hard · 5 marks

Question 6 · hard · 4 marks

Question 7 · hard · 5 marks

Common angles and polygons mistakes

Angles and Polygons FAQ

Next steps

Practise angles and polygons with your progress saved

More Unit 5 topics: Geometry