Choose the correct arc for center and circumference angles
Relate center and circumference angles on the same intercepted arc · Use supplementary opposite cyclic angles · Handle a reflex central arc correctly
Exam scope and assessment notes
Extended content; not a Core requirement. 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
- Core circle properties
- Interior/exterior angle reasoning
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
- Relate center and circumference angles on the same intercepted arc
- Use supplementary opposite cyclic angles
- Handle a reflex central arc correctly
Identify the intercepted arc
For chord AB and circumference vertex C, angle ACB intercepts the arc AB that does not contain C. The central angle standing on that same arc is twice angle ACB. If C lies on the major arc, the intercepted arc is minor; if C lies on the minor arc, the relevant central turn is reflex.
The pictured minor central angle is 112° and C is on the major arc, so the circumference angle is 56°. Do not automatically halve the smaller angle at the center when C is on the other arc.
Cyclic quadrilaterals
In a convex quadrilateral whose four vertices lie on a circle in their boundary order, opposite interior angles sum to 180°. The two intercepted arcs complete 360°, and their half-angles therefore complete 180°.
Angles subtended by one chord from vertices on the same arc are equal; from opposite arcs they are supplementary. State the arc or cyclic-quadrilateral condition in the explanation.
Sources for this lesson
Pause and explain
If C lies on the minor arc AB, which arc does angle ACB intercept?
Worked example
The minor central angle AOB is 112° and C lies on the major arc AB. Find angle ACB.
Show the worked solution
- C's angle intercepts the minor arc AB.
- The central angle on that arc is twice the circumference angle.
Answer 56°.
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 central angle on the same intercepted arc as a circumference angle is 134°. Find the circumference angle.
Give me a hint
Use the two-to-one relationship on the same arc.
Compare my reasoning
- 134 = 2θ.
- θ = 67°.
67°.
Look for these in your work
- Match the intercepted arc
- Divide the central angle by two
A circumference angle is 58°. Find the central angle on its intercepted arc. Separately, find the angle opposite 73° in a convex cyclic quadrilateral.
Give me a hint
Use separate theorems for the two parts.
Compare my reasoning
- Central angle = 2×58 = 116°.
- Opposite cyclic angle = 180−73 = 107°.
116°; opposite cyclic angle 107°.
Look for these in your work
- Choose the correct relationship
- Identify opposite vertices
The minor central angle AOB is 110° and C lies on that minor arc. Find angle ACB.
Give me a hint
C's angle uses the other, major arc.
Compare my reasoning
- Relevant central turn = 360−110 = 250°.
- ACB = 250/2 = 125°.
125°, using the major intercepted arc.
Look for these in your work
- Use the vertex's position
- Do not blindly halve 110°
Common mistakes
- Choose the arc not containing the circumference vertex.
- Opposite, not arbitrary adjacent, cyclic angles are supplementary.
What can you explain now?
If C lies on the minor arc AB, which arc does angle ACB intercept?
Compare with the explanation
The major arc AB, not containing C.
The corresponding central angle is then the reflex turn.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
If C lies on the minor arc AB, which arc does angle ACB intercept?
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.