Central angles, cyclic shapes and tangent chords
Use the remaining Extended circle-angle theorems, matching the same chord and arc carefully.
Section Paper 4 · Extended · Calculator
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Use the two Core circle theorems.
- Identify the arc subtended by a chord.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Half of 112° is what?
What you will learn
- Use the centre-angle and same-segment theorems.
- Find opposite angles in a cyclic quadrilateral.
- Use the alternate segment theorem for a tangent and chord.
Match the chord and the intercepted arc
The angle at the centre is twice an angle at the circumference subtended by the same arc. For chord AB, a minor central angle AOB of 112° corresponds to angle ACB = 56° when C lies on the opposite, major arc. The diagram marks this placement. If C moves onto the minor arc, its intercepted arc changes; you cannot automatically halve the same minor central angle.
Angles subtended by the same chord in the same segment are equal. If C and D both lie on the same side of chord AB on the circumference, angles ACB and ADB match. Being on the same circle alone is insufficient: identify the common chord and segment. In questions that use a reflex central angle, apply the doubling relationship to the arc actually intercepted by the circumference angle.
Opposite angles of a cyclic quadrilateral
A cyclic quadrilateral has all four vertices on one circle. Its opposite angles total 180°. If ABCD is cyclic and angle ABC = 103°, the opposite angle ADC is 77°. Adjacent angles are not generally supplementary by this theorem; check the vertex order before subtracting.
An exterior angle formed by extending one side equals the opposite interior angle of a cyclic quadrilateral: the exterior is supplementary to its adjacent interior angle, and that interior is also supplementary to the opposite angle. You can derive this relationship from two 180° statements rather than memorising an unexplained extra rule.
A tangent and a chord look to the opposite arc
The alternate segment theorem says the angle between a tangent and a chord at contact equals the angle subtended by that chord at the circumference in the alternate segment. If the angle between tangent AT and chord AB is 48°, then the corresponding angle ACB on the opposite arc is 48°. Specify the chosen tangent ray and segment because the other angle at contact is supplementary.
This is a different rule from tangent–radius perpendicularity. A chord is not generally a radius, so its angle with the tangent need not be 90°. In a multi-step problem, mark the tangent–chord angle, match it to the circumference angle, then use a triangle sum or same-segment equality as needed. Give the name 'alternate segment theorem' with the matched chord in your reasoning.
Pause and explain
ABCD is cyclic. Angle ABC is 103°. Find ADC.
Worked example
O is the centre, minor angle AOB = 112°, and C and D are on the major arc AB. Find ACB and ADB.
Show the worked solution
- Both circumference angles intercept minor arc AB.
- The centre-angle theorem gives ACB = 112/2 = 56°.
- ADB = ACB = 56° because they subtend chord AB in the same segment.
Answer ACB = ADB = 56°.
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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 minor central angle AOB is 146°. C and D lie on the major arc AB. Find ACB and ADB.
Give me a hint
Both angles intercept the same minor arc.
Compare my reasoning
- The angle at C is half the central angle: 146/2 = 73°.
- The angle at D subtends the same chord in the same segment.
- ADB is therefore also 73°; state the two theorem reasons.
ACB = ADB = 73°.
Look for these in your work
- Matched the arc and segment.
- Used centre doubling and same-segment equality correctly.
ABCD is cyclic, with angle A = 82° and angle B = 107°. Find angles C and D.
Give me a hint
Opposite pairs are A with C and B with D.
Compare my reasoning
- C = 180 − 82 = 98°.
- D = 180 − 107 = 73°.
- Each opposite pair sums to 180°, and all four angles total 360°.
C = 98°; D = 73°.
Look for these in your work
- Matched opposite, not adjacent, vertices.
- Gave the cyclic-quadrilateral reason.
A tangent at A makes a 48° angle with chord AB. C is on the opposite arc so ACB is the alternate-segment angle. Angle ABC = 61°. Find ACB and BAC.
Give me a hint
Use the tangent–chord theorem first, then the triangle total.
Compare my reasoning
- ACB = 48° by the alternate segment theorem.
- BAC = 180 − 48 − 61.
- BAC = 71°; the tangent–radius right-angle rule is not the rule for chord AB.
ACB = 48°; BAC = 71°.
Look for these in your work
- Used the chosen tangent ray and opposite segment.
- Combined the circle theorem with a triangle sum.
Common mistakes
- Halving a central angle that intercepts a different arc.
- Making adjacent cyclic angles supplementary without another reason.
- Using the tangent–radius rule for a tangent–chord angle.
What can you explain now?
A circumference angle is 37°. What is the central angle subtending the same arc?
Compare with the explanation
74°
The angle at the centre is twice the matching circumference angle: 2 × 37 = 74°.
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Make it stick
Tomorrow, sketch one example of each of the four Extended angle theorems. Mark the common chord, arc or opposite vertices explicitly.
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.