Use circle angle theorems I
Find angles using the semicircle, tangent-radius, centre-circumference, same-segment, cyclic-quadrilateral and alternate-segment properties.
Section 5 · Geometry
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 angle is subtended by a diameter at the circumference?
What you will learn
- Use the angle in a semicircle and radius-tangent right angle
- Apply Extended theorems for centres, segments and cyclic quadrilaterals
- Explain which circle property supports each result
Core starts with two right-angle facts
An angle subtended by a diameter at the circumference is 90°. A radius meets a tangent at 90° at the contact point.
Identify the diameter or point of contact before using the theorem; these are geometric reasons rather than visual estimates.
Extended adds linked circle angles
The angle at the centre is twice the angle at the circumference standing on the same arc. Angles in the same segment are equal; opposite angles of a cyclic quadrilateral total 180°.
The alternate segment theorem links the angle between a tangent and chord to the angle in the opposite segment. Name the chord or arc that matches the angles.
Pause and explain
A centre angle is 100° on an arc. What is the circumference angle on the same arc?
Worked example
A centre angle standing on an arc is 130°. Find the circumference angle on the same arc.
Show the worked solution
- The centre angle is twice the circumference angle.
- Circumference angle = 130°/2.
- This gives 65°.
Answer 65°.
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 cyclic quadrilateral has one angle of 112°. Find its opposite angle.
Give me a hint
Opposite angles in a cyclic quadrilateral total 180°.
Compare my reasoning
- Let the opposite angle be x.
- x + 112° = 180°.
- x = 68°.
68°.
Look for these in your work
- Cyclic quadrilateral identified
- Supplementary relation used
- Angle calculated
A tangent touches a circle at T. The angle between tangent and chord TA is 48°. Find the angle in the alternate segment subtended by TA.
Give me a hint
Use the alternate segment theorem.
Compare my reasoning
- The tangent-chord angle corresponds to the circumference angle in the opposite segment.
- Those angles are equal.
- The required angle is 48°.
48°.
Look for these in your work
- Chord identified
- Opposite segment selected
- Equality stated
A circular window has diameter AB and point C on its circumference. If angle CAB is 31°, find angles ACB and ABC with reasons.
Give me a hint
Use the diameter theorem, then the triangle angle sum.
Compare my reasoning
- Angle ACB = 90° because AB is a diameter.
- Angles in triangle ABC total 180°.
- Angle ABC = 180° − 90° − 31° = 59°.
Angle ACB 90°; angle ABC 59°.
Look for these in your work
- Diameter theorem applied
- Triangle sum used
- Requested angles identified
Common mistakes
- Doubling the centre angle instead of halving it
- Using same-segment equality on different arcs
- Forgetting opposite cyclic angles total 180°
- Applying the alternate-segment rule without matching the chord
What can you explain now?
What is the angle between a radius and tangent at contact?
Compare with the explanation
90°
The radius is perpendicular to the tangent.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do the diameter, tangent, centre, cyclic-quadrilateral and alternate-segment theorems differ?
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.