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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.

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  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?

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    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.

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Flashcards 16 cards to flip through Exam questions 6 questions, 21 marks with answers
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