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Exam questions · Maths · Circle Theorems

The Alternate Segment Theorem

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

    The diagram is not drawn to scale. \(TAS\) is a tangent to the circle at \(A\). \(B\) and \(C\) are points on the circle. Angle \(SAB = 62^\circ\). Calculate angle \(ACB\), giving 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 \(ACB = 62^\circ\), because the angle between a tangent and a chord equals the angle in the alternate segment.

    Mark scheme

    • \(62\) — B1
    • The angle between a tangent and a chord equals the angle in the alternate segment — B1
  2. 2 Calculate [4 marks]

    The diagram is not drawn to scale. \(TAS\) is a tangent to the circle at \(A\). \(B\) and \(C\) are points on the circle. Angle \(TAB = 38^\circ\) and angle \(ABC = 55^\circ\). Calculate angle \(BAC\), giving reasons for your answer. [4 marks]

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

    Angle \(ACB = 38^\circ\), because the angle between a tangent and a chord equals the angle in the alternate segment. Then \(BAC = 180 - 38 - 55 = 87^\circ\).

    Mark scheme

    • \(ACB = 38\) — B1
    • \(180 - 38 - 55\) — M1
    • \(87\) — A1
    • Alternate segment theorem, and the angles in a triangle add up to 180 degrees — B1
  3. 3 Calculate [3 marks]

    \(TA\) is a tangent to a circle at \(A\), and \(B\) and \(C\) are points on the circle, with \(C\) in the alternate segment. Angle \(TAB = 3x + 4\) and angle \(ACB = 5x - 22\). Calculate the value of \(x\). [3 marks]

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    Model answer

    By the alternate segment theorem, \(3x + 4 = 5x - 22\). Then \(26 = 2x\), so \(x = 13\).

    Mark scheme

    • \(3x + 4 = 5x - 22\) — M1
    • \(2x = 26\) — M1
    • \(13\) — A1
  4. 4 Calculate [4 marks]

    The diagram is not drawn to scale. \(PA\) and \(PB\) are tangents to a circle, centre \(O\). \(C\) is a point on the circle. Angle \(APB = 76^\circ\). Calculate angle \(ACB\), giving 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 \(PAB = (180 - 76) \div 2 = 52^\circ\). By the alternate segment theorem, \(ACB = PAB = 52^\circ\).

    Mark scheme

    • \((180 - 76) \div 2\) — M1
    • \(PAB = 52\) — A1
    • \(ACB = 52\) — B1
    • Alternate segment theorem stated — B1
  5. 5 Calculate [4 marks]

    \(AD\) is a diameter of a circle. \(TA\) is a tangent to the circle at \(A\). \(B\) is a point on the circle. Angle \(TAB = 29^\circ\). (a) Calculate angle \(ADB\), giving a reason for your answer. [2 marks] (b) Calculate angle \(DAB\). [2 marks]

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    Model answer

    (a) \(ADB = 29^\circ\), by the alternate segment theorem. (b) \(TAD = 90^\circ\), because a tangent is perpendicular to the radius, so \(DAB = 90 - 29 = 61^\circ\).

    Mark scheme

    • (a) \(29\) — B1
    • (a) Alternate segment theorem stated — B1
    • (b) \(90 - 29\), using angle \(TAD = 90^\circ\) — M1
    • (b) \(61\) — A1
  6. 6 Prove [4 marks]

    \(TA\) 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]

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    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 \(TAB = ADB\).

    Mark scheme

    • 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 — B1

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.