Calculate angles in polygons and parallel lines
Use angle facts with clear reasons, including triangle, quadrilateral, parallel-line and regular-polygon properties.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the angle on a straight line?
What you will learn
- Use angles at a point, on a line and in triangles or quadrilaterals
- Apply corresponding, alternate and co-interior angle facts
- Find polygon interior and exterior angles
State the fact as well as the calculation
Angles at a point total 360°; a straight line is 180°; vertically opposite angles are equal. Triangle and quadrilateral interior sums are 180° and 360°.
For parallel lines, corresponding and alternate angles are equal; co-interior angles total 180°. Use three-letter notation such as angle ABC and name the geometric reason.
Polygon sums follow from triangles
An n-sided polygon has interior sum (n − 2) × 180°. For a regular polygon, divide by n; each exterior angle is 360°/n.
Extended includes irregular polygons too. The exterior turn at each vertex totals 360° around a closed polygon.
Pause and explain
Two co-interior angles are between parallel lines. What is their sum?
Worked example
Find each interior angle of a regular hexagon.
Show the worked solution
- Interior sum = (6 − 2) × 180° = 720°.
- All six angles are equal.
- Each is 720°/6 = 120°.
Answer 120°.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A triangle has angles 48° and 67°. Find the third and state the reason.
Give me a hint
Use the triangle interior-angle sum.
Compare my reasoning
- Known angles total 115°.
- 180° − 115° = 65°.
- Angles in a triangle sum to 180°.
65°.
Look for these in your work
- Known angles added
- Triangle total applied
- Reason stated
Find each interior and exterior angle of a regular decagon.
Give me a hint
Use the interior sum formula, then divide equally; one exterior turn is 360°/n.
Compare my reasoning
- Interior sum = (10 − 2) × 180° = 1440°.
- Each interior angle = 1440°/10 = 144°.
- Each exterior angle = 360°/10 = 36°.
Interior 144°; exterior 36°.
Look for these in your work
- Interior formula used
- Regularity used
- Exterior angle calculated
Two parallel road edges are crossed by a path. One acute angle is 62°. Find its alternate angle and the adjacent co-interior angle, with reasons.
Give me a hint
Alternate angles are equal; co-interior angles on the same side sum to 180°.
Compare my reasoning
- The alternate angle is 62°.
- The co-interior angle is 180° − 62° = 118°.
- Both relationships follow because the road edges are parallel.
Alternate 62°; co-interior 118°.
Look for these in your work
- Parallel condition used
- Equality named
- Supplementary pair calculated
Common mistakes
- Using the exterior sum as the interior sum
- Assuming co-interior angles are equal
- Giving a number without a reason
- Using the wrong vertex as the middle letter in an angle name
What can you explain now?
What is the interior-angle sum of a quadrilateral?
Compare with the explanation
360°
A quadrilateral's angles total 360°.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Which formula gives an n-sided polygon's interior sum, and how do parallel lines relate angles?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.