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AQA MathsGeometry and measures

Angles, lines and polygons

Use angle facts in lines, triangles and polygons.

AQAGCSE MathsGeometry and measuresFoundation and Higher
Visual model

Angle facts on parallel lines

corresponding anglessame position
Mark parallel lines first.
Measure angles from the line or vertex shown.
Name the angle rule before calculating.
Gold-standard guide
20 mins

What you will learn

Use angle facts in lines, triangles and polygons.
Use a clear step-by-step method for angles, lines and polygons.
Check your answer and avoid the most common exam mistake.
Useful before you start
Core number skillsEarlier geometry and measures skillsShowing clear working
Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Method

Sum of interior angles of an n-sided polygon = (n − 2) × 180°

Step 1

Use the interior angle sum formula for n = 8

Sum = (8 − 2) × 180° = 6 × 180° = 1080°

Step 2

Divide by the number of sides for a regular polygon

One interior angle = 1080° ÷ 8 = 135°

Step 3

Verify using exterior angles

Exterior angle = 360°/8 = 45°

Watch out

Watch out

Students use (n − 2) × 180° correctly but then forget to divide by n for the regular polygon

f
Triangle

angles in a triangle sum to 180 degrees.

f
Polygon exterior

exterior angles of any polygon sum to 360 degrees.

Worked example

Find the size of one interior angle of a regular octagon.

1

Use the interior angle sum formula for n = 8: Sum = (8 − 2) × 180° = 6 × 180° = 1080°.

2

Divide by the number of sides for a regular polygon: One interior angle = 1080° ÷ 8 = 135°.

3

Verify using exterior angles: Exterior angle = 360°/8 = 45°. Interior = 180° − 45° = 135°. ✓

Final answer

135°

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Find the size of one interior angle of a regular octagon.

3 marks4 minsangles-lines-and-polygons-worked
Show solution
Worked solution
  1. 1.Use the interior angle sum formula for n = 8: Sum = (8 − 2) × 180° = 6 × 180° = 1080°.
  2. 2.Divide by the number of sides for a regular polygon: One interior angle = 1080° ÷ 8 = 135°.
  3. 3.Verify using exterior angles: Exterior angle = 360°/8 = 45°. Interior = 180° − 45° = 135°. ✓
Final answer

135°

Mark points
  • M1: use the interior angle sum formula for n = 8
  • M1: divide by the number of sides for a regular polygon
  • M1: verify using exterior angles
  • A1: 135°
Watch out

Students use (n − 2) × 180° correctly but then forget to divide by n for the regular polygon.The formula gives the sum of all angles — dividing gives each individual angle.Always read whether the question asks for the total or one angle.

Diagnosticrecall

Find the sum of interior angles of a 10-sided polygon.

1 mark2 minsangles-lines-and-polygons-q1
Show solution
Worked solution
  1. 1.Spot the skill: Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
  2. 2.Use the use the interior angle sum formula for n = 8 stage first, then divide by the number of sides for a regular polygon.
  3. 3.Keep the final answer visible: 1440°.
Final answer

1440°

Mark points
  • M1: use the correct sum of interior angles of an n-sided polygon = (n − 2) × 180°.for a regular polygon, all angles are equal, so divide by n. exterior angle of any polygon = 360°/n.interior + exterior = 180°.
  • A1: 1440°
Watch out

Students use (n − 2) × 180° correctly but then forget to divide by n for the regular polygon.The formula gives the sum of all angles — dividing gives each individual angle.Always read whether the question asks for the total or one angle.

Easyprocedure

The exterior angle of a regular polygon is 24°. How many sides?

2 marks3 minsangles-lines-and-polygons-q2
Show solution
Worked solution
  1. 1.Spot the skill: Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
  2. 2.Use the divide by the number of sides for a regular polygon stage first, then verify using exterior angles.
  3. 3.Keep the final answer visible: 15.
Final answer

15

Mark points
  • M1: use the correct sum of interior angles of an n-sided polygon = (n − 2) × 180°.for a regular polygon, all angles are equal, so divide by n. exterior angle of any polygon = 360°/n.interior + exterior = 180°.
  • A1: 15
Watch out

Students use (n − 2) × 180° correctly but then forget to divide by n for the regular polygon.The formula gives the sum of all angles — dividing gives each individual angle.Always read whether the question asks for the total or one angle.

Mediumreasoning

Find the value of x if two angles on a straight line are 3x + 10 and 5x + 26.

3 marks4 minsangles-lines-and-polygons-q3
Show solution
Worked solution
  1. 1.Spot the skill: Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
  2. 2.Use the verify using exterior angles stage first, then use the interior angle sum formula for n = 8.
  3. 3.Keep the final answer visible: x = 18°.
Final answer

x = 18°

Mark points
  • M1: use the correct sum of interior angles of an n-sided polygon = (n − 2) × 180°.for a regular polygon, all angles are equal, so divide by n. exterior angle of any polygon = 360°/n.interior + exterior = 180°.
  • A1: x = 18°
Watch out

Students use (n − 2) × 180° correctly but then forget to divide by n for the regular polygon.The formula gives the sum of all angles — dividing gives each individual angle.Always read whether the question asks for the total or one angle.

Hardproblem solving

In a triangle, two angles are 52° and 67°. Find the third angle.

3 marks5 minsangles-lines-and-polygons-q4
Show solution
Worked solution
  1. 1.Spot the skill: Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
  2. 2.Use the use the interior angle sum formula for n = 8 stage first, then divide by the number of sides for a regular polygon.
  3. 3.Keep the final answer visible: 61°.
Final answer

61°

Mark points
  • M1: use the correct sum of interior angles of an n-sided polygon = (n − 2) × 180°.for a regular polygon, all angles are equal, so divide by n. exterior angle of any polygon = 360°/n.interior + exterior = 180°.
  • A1: 61°
Watch out

Students use (n − 2) × 180° correctly but then forget to divide by n for the regular polygon.The formula gives the sum of all angles — dividing gives each individual angle.Always read whether the question asks for the total or one angle.

Exam-stylemulti-step

Angles in a quadrilateral are 85°, 95°, 110° and x°. Find x.

4 marks6 minsangles-lines-and-polygons-q5
Show solution
Worked solution
  1. 1.Spot the skill: Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
  2. 2.Use the divide by the number of sides for a regular polygon stage first, then verify using exterior angles.
  3. 3.Keep the final answer visible: 70°.
Final answer

70°

Mark points
  • M1: use the correct sum of interior angles of an n-sided polygon = (n − 2) × 180°.for a regular polygon, all angles are equal, so divide by n. exterior angle of any polygon = 360°/n.interior + exterior = 180°.
  • A1: 70°
Watch out

Students use (n − 2) × 180° correctly but then forget to divide by n for the regular polygon.The formula gives the sum of all angles — dividing gives each individual angle.Always read whether the question asks for the total or one angle.

Grade 9 stretchproblem solving

A regular polygon has exterior angle 24 degrees. Find the number of sides.

4 marks7 minspolygon-g9
Show solution
Worked solution
  1. 1.Exterior angles sum to 360 degrees.
  2. 2.Divide 360 by one exterior angle.
Final answer

15 sides

Mark points
  • M1: use 360 / 24
  • A1: 15
Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint
12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Angles, lines and polygons - 2 marksFind the sum of interior angles of a 10-sided polygon.Mark answer
Answer

1440°

2Properties of shapes - 2 marksHow many lines of symmetry does a regular pentagon have?Mark answer
Answer

5

3Perimeter, area and volume - 2 marksA prism has cross-section area 24 cm2 and length 7 cm. Find its volume.Mark answer
Answer

168 cm3

4Circles - 3 marksFind the diameter of a circle with area 100pic100pi cm2.Mark answer
Answer

20 cm

Mastery check
  • I can explain the method for angles, lines and polygons.
  • I can show clear working without skipping key steps.
  • I can avoid this mistake: Students use (n − 2) × 180° correctly but then forget to divide by n for the regular polygon.The formula gives the sum of all angles — dividing gives each individual angle.Always read whether the question asks for the total or one angle.
Ask Nova Bot
Explain angles, lines and polygons from the beginning.Give me a hint for a angles, lines and polygons question.Quiz me on angles, lines and polygons.
Related topics
Properties of shapesPerimeter, area and volumeCircles
Official exam-board sources

This guide follows the AQA GCSE Mathematics 8300 specification. Practice questions are original Learnova questions shaped around official content and exam skills.

Official specification Official past papers
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