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Exam questions · Maths · Circle Theorems

The Alternate Segment Theorem

  • 6 exam questions
  • 21 marks
  • 9 quick checks
  1. 1 Work out [2 marks]

    Not drawn accurately. \(TAS\) is a tangent to the circle at \(A\). \(B\) and \(C\) are points on the circumference. Angle \(SAB = 35^\circ\). 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.
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    Model answer

    Angle \(ACB = 35^\circ\), because the angle between a tangent and a chord equals the angle in the alternate segment.

    Mark scheme

    • \(35\) — B1
    • The angle between a tangent and a chord equals the angle in the alternate segment — Q1
  2. 2 Work out [4 marks]

    Not drawn accurately. \(TAS\) is a tangent to the circle at \(A\). \(B\) and \(C\) are points on the circumference. Angle \(TAB = 46^\circ\) and angle \(ABC = 60^\circ\). Work out the size of angle \(BAC\). Give reasons for your answer. [4 marks]

    A circle diagram showing a tangent and a triangle ABC.
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    Model answer

    Angle \(ACB = 46^\circ\), because the angle between a tangent and a chord equals the angle in the alternate segment. Then \(BAC = 180 - 46 - 60 = 74^\circ\).

    Mark scheme

    • \(ACB = 46\) — B1
    • \(180 - 46 - 60\) — M1
    • \(74\) — A1
    • Alternate segment theorem, and the angles in a triangle add up to 180 degrees — Q1
  3. 3 Work out [3 marks]

    \(TA\) is a tangent to a circle at \(A\). \(B\) and \(C\) are points on the circumference, with \(C\) in the alternate segment. Angle \(TAB = 2x + 18\) and angle \(ACB = 4x - 12\). Work out the value of \(x\). [3 marks]

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    Model answer

    By the alternate segment theorem, \(2x + 18 = 4x - 12\). Then \(30 = 2x\), so \(x = 15\).

    Mark scheme

    • \(2x + 18 = 4x - 12\) — M1
    • \(2x = 30\) — M1
    • \(15\) — A1
  4. 4 Work out [4 marks]

    Not drawn accurately. \(PA\) and \(PB\) are tangents to a circle, centre \(O\). \(C\) is a point on the circumference. Angle \(APB = 68^\circ\). 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.
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    Model answer

    \(PA = PB\), so triangle \(PAB\) is isosceles and \(PAB = (180 - 68) \div 2 = 56^\circ\). By the alternate segment theorem, \(ACB = PAB = 56^\circ\).

    Mark scheme

    • \((180 - 68) \div 2\) — M1
    • \(PAB = 56\) — A1
    • \(ACB = 56\) — B1
    • Alternate segment theorem stated — Q1
  5. 5 Work out [4 marks]

    \(AD\) is a diameter of a circle. \(TA\) is a tangent at \(A\). \(B\) is a point on the circumference. Angle \(TAB = 33^\circ\). (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]

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    Model answer

    (a) \(ADB = 33^\circ\), by the alternate segment theorem. (b) \(TAD = 90^\circ\), because a tangent is perpendicular to the radius, so \(DAB = 90 - 33 = 57^\circ\).

    Mark scheme

    • (a) \(33\) — B1
    • (a) Alternate segment theorem stated — Q1
    • (b) \(90 - 33\), using angle \(TAD = 90^\circ\) — M1
    • (b) \(57\) — A1
  6. 6 Prove [4 marks]

    \(TA\) is a tangent at \(A\) to a circle, centre \(O\). \(B\) and \(C\) are points on the circumference, with \(C\) in the alternate segment. Use the circle theorems for the centre and for tangents to prove that angle \(TAB\) equals angle \(ACB\). [4 marks]

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    Model answer

    \(OA\) is perpendicular to \(TA\), so \(OAB = 90^\circ - TAB\). \(OA = OB\), so \(OBA = OAB\) and \(AOB = 180 - 2(90 - TAB) = 2 \times TAB\). The angle at the centre is twice the angle at the circumference, so \(ACB = \dfrac{1}{2}AOB = TAB\).

    Mark scheme

    • \(OAB = 90^\circ - TAB\), because the tangent is perpendicular to the radius — B1
    • Triangle \(OAB\) is isosceles, so \(OBA = OAB\) — B1
    • \(AOB = 180 - 2(90 - TAB) = 2 \times TAB\) — M1
    • \(ACB = \dfrac{1}{2}AOB = TAB\), as the angle at the centre is twice the angle at the circumference — Q1

Quick check

  1. 1

    What does the alternate segment theorem say?

    1. AThe angle between a tangent and a chord equals the angle in the alternate segment
    2. BThe angle between a tangent and a chord is \(90^\circ\)
    3. CAngles in the same segment are equal
    4. DThe angle in a semicircle is \(90^\circ\)
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    A: The angle between a tangent and a chord equals the angle in the alternate segment

    This is the alternate segment theorem.

  2. 2

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

    1. AAt the centre of the circle
    2. BAnywhere on the circle
    3. CAt the end of a diameter
    4. DAt the point of contact of the tangent
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    D: At the point of contact of the tangent

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

  3. 3

    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\(58^\circ\)
    4. D\(122^\circ\)
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    C: \(58^\circ\)

    They are equal.

  4. 4

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

    1. AOpposite the centre
    2. BOn the other side of the chord
    3. CEvery other segment
    4. DInside the triangle
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    B: On the other side of the chord

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

  5. 5

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

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

  6. 6

    \(PA\) 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\(58^\circ\)
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    D: \(58^\circ\)

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

  7. 7

    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\(14\)
    4. D\(7\)
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    C: \(14\)

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

  8. 8

    \(AD\) 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\(54^\circ\)
    3. C\(90^\circ\)
    4. D\(144^\circ\)
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    B: \(54^\circ\)

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

  9. 9

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

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