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

IGCSE Maths Angles and Polygons Questions and Answers

Angles and Polygons questions with full solutions for Pearson Edexcel International GCSE Mathematics A (4MA1), Higher tier. Read the 6 worked examples first, then try the 3 exam-style questions yourself and check your working against the mark scheme.

Practise angles and polygons in Unit 5 →Angles and Polygons revision notes

What this topic covers

Angles and Polygons worked examples

Worked example 1: Angles on a straight line

Angles of \(3x^\circ\), \(2x^\circ\) and \(40^\circ\) lie on a straight line. Find \(x\).

Solution

  1. Angles on a straight line add to \(180^\circ\): \(5x+40=180\).
  2. \(x=28\).

Answer: \(x=28\)

Tip: Always state the angle fact you use.

Worked example 2: Co-interior angles

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

Solution

  1. Co-interior angles add to \(180^\circ\): \(3x+30=180\), \(x=50\).
  2. Angles: \(70^\circ\) and \(110^\circ\).

Answer: 70° and 110°

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

Worked example 3: Isosceles triangle

An isosceles triangle has an apex angle of \(38^\circ\). Find each base angle.

Solution

  1. The base angles are equal: \((180-38)\div2\).
  2. \(=71^\circ\).

Answer: 71°

Tip: Identify which angle is the 'odd one out'.

Worked example 4: Exterior angle of a triangle

Two interior angles of a triangle are \(57^\circ\) and \(68^\circ\). Find the exterior angle at the third vertex.

Solution

  1. The exterior angle equals the sum of the two interior opposite angles.
  2. \(57+68=125^\circ\).

Answer: 125°

Tip: Check: third angle 55°, and 180 − 55 = 125.

Worked example 5: Interior angle of a regular polygon

Find each interior angle of a regular octagon.

Solution

  1. Exterior angle \(=360\div8=45^\circ\).
  2. Interior \(=180-45=135^\circ\).

Answer: 135°

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

Worked example 6: Missing angle in a hexagon

A hexagon has angles 100°, 130°, 125°, 140°, 95° and \(x^\circ\). Find \(x\).

Solution

  1. Angle sum \(=(6-2)\times180=720^\circ\).
  2. \(x=720-590=130\).

Answer: 130°

Tip: Sum of interior angles = (n − 2) × 180°.

Angles and Polygons exam-style questions

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

Question 1 · 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 \( BPQ = (5x - 12)^\circ \) and angle \( PQC = (3x + 24)^\circ \). (a) Find the value of \( x \). (b) Work out the size of angle APQ. Give reasons for your answers.

Show the answer and mark scheme
  • 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°

Question 2 · medium · 4 marks

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

Show the answer and 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°

Question 3 · 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 \( QBC = (2y + 10)^\circ \), angle \( SDC = (3y - 5)^\circ \) and angle \( BCD = 110^\circ \) (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°

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

Revise angles and polygons for the exam

Questions students ask

Which Edexcel IGCSE Maths papers test angles and polygons?

Either Higher tier paper, 1H or 2H, can test angles and polygons: both cover the whole Pearson Edexcel International GCSE Mathematics A (4MA1) specification and both allow a calculator. Show your method, because most marks are for working.

Where can I practise more angles and polygons questions?

In the IGCSE student hub: Unit 5 (Geometry) has more exam-style questions on this topic, each with a mark scheme. Practice starts with a 7-day free trial; the worked examples and questions on this page are free.

More Unit 5 topics: Geometry