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Maths · Circle Theorems

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The Alternate Segment Theorem

Finding angles between a tangent and a chord using the alternate segment theorem.

  • Higher
  • 9 key terms
  • All boards

Learning Objectives

  1. 1State the alternate segment theorem.
  2. 2Find the alternate segment from a diagram.
  3. 3Use the theorem to find angles between a tangent and a chord.
  4. 4Combine it with other circle theorems and give full reasons.

The angle between a tangent and a chord

When a chord is drawn from the point where a tangent touches a circle, the angle between the tangent and the chord is linked to an angle inside the circle. The link is called the alternate segment theorem. "Alternate" means the other side: the angle in the segment on the opposite side of the chord. This theorem is often the hardest to spot, so it is worth learning to recognise the pattern of a tangent, a chord from the point of contact, and a triangle in the opposite segment.

The theorem

The angle between a tangent and a chord equals the angle in the alternate segment.

  • The theorem

    The angle between a tangent and a chord equals the angle in the alternate segment.

  • The tangent angle

    The angle is made by the tangent and the chord, at the point of contact.

  • The alternate segment

    The segment on the other side of the chord from that angle.

  • The reason

    "Angle between tangent and chord equals angle in the alternate segment."

Using the theorem

\(TA\) is a tangent to a circle at \(A\). \(B\) and \(C\) are points on the circle. Angle \(TAB = 62^\circ\). Work out angle \(ACB\), where \(C\) is in the alternate segment, and give a reason.

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  1. 1 Spot the pattern \(TA\) is a tangent and \(AB\) is a chord from the point of contact.
  2. 2 Alternate segment \(C\) is on the other side of \(AB\) from angle \(TAB\).
  3. 3 Apply the theorem Angle \(ACB\) equals angle \(TAB\).
  4. 4 Reason "Angle between tangent and chord equals angle in the alternate segment", so \(ACB = 62^\circ\).

AnswerAngle \(ACB = 62^\circ\)

Combining theorems

Most exam questions use the alternate segment theorem with another theorem.

  • Triangles

    Add the angles of a triangle to \(180^\circ\) to find the missing angle.

  • Isosceles triangle

    Two radii make an isosceles triangle, so the base angles are equal.

  • Tangent and radius

    The radius to the point of contact is perpendicular to the tangent.

  • One step at a time

    Find each angle, write its reason, then use it in the next step.

Two theorems together

\(TA\) is a tangent to a circle at \(A\). \(B\) and \(C\) are points on the circle. Angle \(TAB = 48^\circ\) and angle \(ABC = 70^\circ\). Work out angle \(BAC\), giving reasons.

Show the solutionHide the solution
  1. 1 First angle Angle \(ACB = 48^\circ\), because the angle between a tangent and a chord equals the angle in the alternate segment.
  2. 2 Triangle The angles in triangle \(ABC\) add up to \(180^\circ\).
  3. 3 Missing angle \(BAC = 180 - 70 - 48 = 62^\circ\).
  4. 4 Reason "Angles in a triangle add up to \(180^\circ\)."

AnswerAngle \(BAC = 62^\circ\)

Test yourself

  1. 1

    State the alternate segment theorem.

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    The angle between a tangent and a chord equals the angle in the alternate segment.

  2. 2

    Where does the chord start?

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    At the point of contact of the tangent.

  3. 3

    Which segment is the alternate segment?

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    The one on the other side of the chord from the angle.

  4. 4

    What else is a tangent perpendicular to?

    Show answerHide answer

    The radius at the point of contact.

  5. 5

    What do the angles of a triangle add up to?

    Show answerHide answer

    \(180^\circ\).

Exam technique: a tangent in the diagram

Whenever you see a tangent, think of three theorems.

  • Radius

    The tangent is perpendicular to the radius at the point of contact.

  • Chord

    A chord from the point of contact gives the alternate segment theorem.

  • Equal tangents

    Two tangents from a point are equal in length.

  • Write the full reason

    "Alternate segment theorem" on its own may not earn the mark, so write it out.

Summary and exam focus

  • The angle between a tangent and a chord equals the angle in the alternate segment.
  • The chord must start at the point of contact.
  • Combine it with the angle sum of a triangle and isosceles triangles.
  • Write the full reason for every angle.

Exam focus

\(TA\) is a tangent to a circle at \(A\). \(B\) and \(C\) are points on the circle. Angle \(TAB = 62^\circ\). Work out angle \(ACB\), and give a reason. (2 marks) (2 marks)

\(ACB = 62^\circ\). The reason is: "The angle between a tangent and a chord equals the angle in the alternate segment."

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Alternate segment
The segment on the other side of a chord from a given angle.
Tangent
A straight line that touches a circle at one point.
Chord
A straight line joining two points on a circle.
Point of contact
The point where a tangent touches the circle.
Segment
The part of a circle cut off by a chord.
Isosceles triangle
A triangle with two equal sides.
Base angles
The two equal angles of an isosceles triangle.
Radius
The distance from the centre to the circle.
Reason
A statement of the theorem that justifies an angle.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Work out 2 marks Core

Diagram NOT accurately drawn. \(TAS\) is a tangent to the circle at \(A\). \(B\) and \(C\) are points on the circle. Angle \(SAB = 48^\circ\). Work out the size of angle \(ACB\). Give a reason for your answer. (2 marks)

A circle diagram showing a tangent, a chord and an angle in the alternate segment.

Mark scheme — 2 marks available

  • \(48\) — B1
  • The angle between a tangent and a chord equals the angle in the alternate segment — C1

Model answer

Angle \(ACB = 48^\circ\), because the angle between a tangent and a chord equals the angle in the alternate segment.

2. Exam question Work out 4 marks Core

Diagram NOT accurately drawn. \(TAS\) is a tangent to the circle at \(A\). \(B\) and \(C\) are points on the circle. Angle \(TAB = 52^\circ\) and angle \(ABC = 65^\circ\). Work out the size of angle \(BAC\). Give reasons for your answer. (4 marks)

A circle diagram showing a tangent and a triangle ABC.

Mark scheme — 4 marks available

  • \(ACB = 52\) — B1
  • Alternate segment theorem stated — C1
  • \(180 - 52 - 65\) — M1
  • \(63\) — A1

Model answer

Angle \(ACB = 52^\circ\), because the angle between a tangent and a chord equals the angle in the alternate segment. Then \(BAC = 180 - 52 - 65 = 63^\circ\), because the angles in a triangle add up to \(180^\circ\).

3. Exam question Work out 3 marks Core

\(TA\) is a tangent to a circle at \(A\), and \(B\) and \(C\) are points on the circle, with \(C\) in the alternate segment. Angle \(TAB = 3x + 10\) and angle \(ACB = 5x - 14\). Work out the value of \(x\). (3 marks)

Mark scheme — 3 marks available

  • \(3x + 10 = 5x - 14\) — M1
  • \(2x = 24\) — M1
  • \(12\) — A1

Model answer

By the alternate segment theorem, \(3x + 10 = 5x - 14\). Then \(24 = 2x\), so \(x = 12\).

4. Exam question Work out 4 marks Core

Diagram NOT accurately drawn. \(PA\) and \(PB\) are tangents to a circle, centre \(O\). \(C\) is a point on the circle. Angle \(APB = 64^\circ\). Work out the size of angle \(ACB\). Give reasons for your answer. (4 marks)

A circle diagram showing two tangents from P and a point C on the circle.

Mark scheme — 4 marks available

  • \((180 - 64) \div 2\) — M1
  • \(PAB = 58\) — A1
  • \(ACB = 58\) — B1
  • Alternate segment theorem stated — C1

Model answer

\(PA = PB\), so triangle \(PAB\) is isosceles and \(PAB = (180 - 64) \div 2 = 58^\circ\). By the alternate segment theorem, \(ACB = PAB = 58^\circ\).

5. Exam question Work out 4 marks Stretch

\(AD\) is a diameter of a circle. \(TA\) is a tangent to the circle at \(A\). \(B\) is a point on the circle. Angle \(TAB = 38^\circ\). (a) Work out the size of angle \(ADB\). Give a reason for your answer. (2 marks) (b) Work out the size of angle \(DAB\). (2 marks)

Mark scheme — 4 marks available

  • (a) \(38\) — B1
  • (a) Alternate segment theorem stated — C1
  • (b) \(90 - 38\), using angle \(TAD = 90^\circ\) — M1
  • (b) \(52\) — A1

Model answer

(a) \(ADB = 38^\circ\), by the alternate segment theorem. (b) \(TAD = 90^\circ\) because a tangent is perpendicular to the radius, so \(DAB = 90 - 38 = 52^\circ\).

6. Exam question Prove 4 marks Stretch

\(TA\) is a tangent to a circle at \(A\). \(AD\) is a diameter. \(B\) is a point on the circle. Prove that angle \(TAB\) equals angle \(ADB\). (4 marks)

Mark scheme — 4 marks available

  • Angle \(TAD = 90^\circ\), because a tangent is perpendicular to the radius — B1
  • Angle \(ABD = 90^\circ\), because the angle in a semicircle is a right angle — B1
  • \(TAB = 90 - BAD\) and \(ADB = 90 - BAD\) — M1
  • Concludes that the angles are equal — C1

Model answer

Angle \(TAD = 90^\circ\), because a tangent is perpendicular to the radius. Angle \(ABD = 90^\circ\), because the angle in a semicircle is \(90^\circ\). So \(TAB = 90 - BAD\), and in triangle \(ABD\), \(ADB = 90 - BAD\). Therefore \(TAB = ADB\).

7. Multiple choice 1 mark Easier

What does the alternate segment theorem say?

  1. A The angle between a tangent and a chord equals the angle in the alternate segment Correct
  2. B The angle between a tangent and a chord is \(90^\circ\)
  3. C Angles in the same segment are equal
  4. D The angle in a semicircle is \(90^\circ\)

Why: This is the alternate segment theorem.

8. Multiple choice 1 mark Easier

Where must the chord start for the alternate segment theorem to apply?

  1. A At the centre of the circle
  2. B Anywhere on the circle
  3. C At the end of a diameter
  4. D At the point of contact of the tangent Correct

Why: The chord starts at the point where the tangent touches the circle.

9. Multiple choice 1 mark Easier

The angle between a tangent and a chord is \(58^\circ\). What is the angle in the alternate segment?

  1. A \(32^\circ\)
  2. B \(116^\circ\)
  3. C \(58^\circ\) Correct
  4. D \(122^\circ\)

Why: They are equal.

10. Multiple choice 1 mark Easier

What does “alternate” mean in the alternate segment theorem?

  1. A Opposite the centre
  2. B On the other side of the chord Correct
  3. C Every other segment
  4. D Inside the triangle

Why: The alternate segment is the one on the opposite side of the chord from the angle.

11. Multiple choice 1 mark Core

\(TA\) is a tangent at \(A\). Angle \(TAB = 40^\circ\) and angle \(ABC = 75^\circ\), where \(C\) is in the alternate segment. What is angle \(BAC\)?

  1. A \(65^\circ\) Correct
  2. B \(40^\circ\)
  3. C \(75^\circ\)
  4. D \(115^\circ\)

Why: \(ACB = 40^\circ\) by the alternate segment theorem, so \(BAC = 180 - 75 - 40 = 65^\circ\).

12. Multiple choice 1 mark Core

\(PA\) and \(PB\) are tangents and \(\angle APB = 64^\circ\). \(C\) is on the major arc. What is angle \(ACB\)?

  1. A \(64^\circ\)
  2. B \(116^\circ\)
  3. C \(32^\circ\)
  4. D \(58^\circ\) Correct

Why: \(PAB = (180 - 64) \div 2 = 58^\circ\), and this equals the angle in the alternate segment.

13. Multiple choice 1 mark Core

The angle between the tangent and the chord is \(2x + 4\) and the angle in the alternate segment is \(3x - 10\). What is \(x\)?

  1. A \(6\)
  2. B \(32\)
  3. C \(14\) Correct
  4. D \(7\)

Why: \(2x + 4 = 3x - 10\), so \(x = 14\).

14. Multiple choice 1 mark Stretch

\(AD\) is a diameter, \(TA\) is a tangent at \(A\) and \(B\) is on the circle. Angle \(TAB = 36^\circ\). What is angle \(DAB\)?

  1. A \(36^\circ\)
  2. B \(54^\circ\) Correct
  3. C \(90^\circ\)
  4. D \(144^\circ\)

Why: \(TAD = 90^\circ\) because a tangent is perpendicular to the radius, so \(DAB = 90 - 36 = 54^\circ\).

15. Multiple choice 1 mark Stretch

A tangent at \(A\) makes an angle of \(x\) with the chord \(AB\). What is the angle \(AOB\) at the centre, on the same side as that angle?

  1. A \(2x\) Correct
  2. B \(x\)
  3. C \(90 - x\)
  4. D \(180 - x\)

Why: The angle in the alternate segment is \(x\), and the angle at the centre is twice that.