Circle Theorems: Angle Properties
The circle theorems are a fixed set of angle facts, and the marks come as much from naming the theorem as from finding the number. The angle at the centre is twice the angle at the circumference subtended by the same arc. The angle in a semicircle is a right angle, which is the special case of that first theorem. Angles in the same segment, subtended by the same arc, are equal. Opposite angles of a cyclic quadrilateral add to 180 degrees. A tangent meets a radius at 90 degrees, and the two tangents from an external point are equal in length. The alternate segment theorem states that the angle between a tangent and a chord equals the angle in the alternate segment.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Mark everything known on the diagram before reasoning: radii equal, right angles at tangents, and any angles given.
- Look for the standard configurations in order: a diameter (giving a right angle at the circumference), a centre angle with a matching circumference angle, a cyclic quadrilateral, and a tangent.
- Use isosceles triangles created by two radii. Two radii are always equal, so the base angles are equal, and that fact unlocks many diagrams that seem to have too little information.
- Write the reason with every step, in the exam's own language: the angle at the centre is twice the angle at the circumference, angles in the same segment are equal, opposite angles in a cyclic quadrilateral sum to 180 degrees, the angle between a tangent and a radius is 90 degrees, or the alternate segment theorem.
- Work in small steps and label each angle you find on the diagram as you go, since later steps almost always use an earlier result.
- Check the total: angles in a triangle sum to 180 degrees and angles around a point to 360 degrees, so use those as a final consistency test.
Worked example
A, B and C are points on a circle with centre O. Angle AOC = 130 degrees, where AOC is the angle at the centre subtended by arc AC not containing B. Find angle ABC, and find angle OAC.
- Angle ABC is at the circumference, subtended by the same arc AC as the centre angle AOC.
- By the theorem that the angle at the centre is twice the angle at the circumference, angle ABC = 130 divided by 2 = 65 degrees.
- For angle OAC, note that OA and OC are both radii, so triangle OAC is isosceles.
- The angles OAC and OCA are therefore equal, and together with the 130 degrees at O they sum to 180 degrees.
- So 2 times angle OAC = 180 - 130 = 50 degrees.
- Therefore angle OAC = 25 degrees. Reasons: angle at the centre is twice the angle at the circumference, and base angles of an isosceles triangle formed by two radii are equal.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1State the theorem about the angle in a semicircle.Show answer
Answer: The angle in a semicircle is 90 degrees, that is the angle subtended by a diameter at the circumference is a right angle.
Q2The angle at the centre subtended by an arc is 84 degrees. Find the angle at the circumference subtended by the same arc.Show answer
Answer: 42 degrees.
Q3In a cyclic quadrilateral, one angle is 108 degrees. Find the angle opposite it.Show answer
Answer: 180 - 108 = 72 degrees.
Q4What is the angle between a tangent and the radius drawn to the point of contact?Show answer
Answer: 90 degrees.
Q5State the alternate segment theorem.Show answer
Answer: The angle between a tangent and a chord equals the angle in the alternate segment, that is the angle subtended by that chord in the segment on the other side.
Q6Why is a triangle formed by two radii always isosceles?Show answer
Answer: Both radii are the same length, so two sides of the triangle are equal, which makes the base angles equal too.
Q7Two tangents are drawn to a circle from the same external point. What can be said about their lengths?Show answer
Answer: They are equal.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
P, Q, R and S lie on a circle. PQRS is a cyclic quadrilateral with angle SPQ = 95 degrees and angle PQR = 71 degrees. Find angle QRS and angle RSP, giving a reason for each answer.
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A, B and C are points on a circle with centre O. AB is a diameter. Angle BAC = 34 degrees. Find angle ABC, giving full reasons.
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Free printable worksheet
Want more practice on paper? Download the circle theorems: angle properties worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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