Use straight, intersecting and parallel-line angle reasons
Use full-turn and straight-line sums · Recognise vertically opposite and parallel-line angle relations · Write a reason for each calculation
Section Paper 3 · Historical Core · calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Angle vocabulary
- Simple equations
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Use full-turn and straight-line sums
- Recognise vertically opposite and parallel-line angle relations
- Write a reason for each calculation
Local angle relationships
Angles around a point sum to 360°; adjacent angles on a straight line sum to 180°. Vertically opposite angles formed by two intersecting lines are equal. State the relationship that connects the particular angles rather than guessing from a picture.
A marked parallel pair
When a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°. These equalities need the parallel condition; similarly placed angles on arbitrary lines need not match.
The diagram's alternate interior angles are 67° and its co-interior partner is 113°. Use concise reasoned arguments. A formal proof format is not required by 0444, but unexplained numbers omit the mathematical justification.
Sources for this lesson
Pause and explain
Are corresponding angles necessarily equal when the two lines are not parallel?
Worked example
A transversal crosses marked parallel lines. One alternate interior angle is 67°. Find its alternate partner and co-interior partner.
Show the worked solution
- Alternate angles on parallel lines are equal, so the partner is 67°.
- Co-interior angles sum to 180°: 180−67 = 113°.
Answer 67° and 113°, with the stated parallel-line reasons.
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.
Two adjacent angles form a straight line; one is 124°. Find the other with a reason.
Give me a hint
Use the straight-line sum.
Compare my reasoning
- The total is 180°.
- 180−124 = 56°.
56°, because adjacent angles on a straight line sum to 180°.
Look for these in your work
- Identify the relationship
- Give a reason
Three angles around a point are 80°, 115° and x. Find x.
Give me a hint
Use a full turn.
Compare my reasoning
- 80+115+x = 360.
- x = 165°.
165°, using the 360° full-turn sum.
Look for these in your work
- Include every sector
- Check the total
Marked parallel lines give a co-interior pair 3x+10° and 2x+20°. Find x and both angles.
Give me a hint
Set their sum to 180°.
Compare my reasoning
- 5x+30 = 180 gives x = 30.
- The angles are 100° and 80°, which sum to 180°.
x = 30; angles 100° and 80°.
Look for these in your work
- Use the parallel condition
- Substitute back to check
Common mistakes
- Parallel-line rules require parallel lines.
- A correct value still needs a reason.
What can you explain now?
Are corresponding angles necessarily equal when the two lines are not parallel?
Compare with the explanation
No; the parallel condition is required.
Position alone does not establish the relationship.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Are corresponding angles necessarily equal when the two lines are not parallel?
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.