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Exam questions · Maths · Circle Theorems

Same Segment and Cyclic Quadrilaterals

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

    Not drawn accurately. \(A\), \(B\), \(C\) and \(D\) are points on the circumference of a circle. Angle \(ACB = 52^\circ\). Work out the size of angle \(ADB\). Give a reason for your answer. [2 marks]

    A circle diagram showing two angles in the same segment.
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    Model answer

    Angle \(ADB = 52^\circ\), because angles in the same segment are equal.

    Mark scheme

    • \(52\) — B1
    • Angles in the same segment are equal — Q1
  2. 2 Work out [4 marks]

    Not drawn accurately. \(ABCD\) is a cyclic quadrilateral. Angle \(BAD = 106^\circ\) and angle \(ABC = 78^\circ\). (a) Work out the size of angle \(BCD\). Give a reason for your answer. [2 marks] (b) Work out the size of angle \(ADC\). [2 marks]

    A circle diagram showing a cyclic quadrilateral with two angles given.
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    Model answer

    (a) \(BCD = 180 - 106 = 74^\circ\), because opposite angles of a cyclic quadrilateral add up to \(180^\circ\). (b) \(ADC = 180 - 78 = 102^\circ\).

    Mark scheme

    • (a) \(74\) — B1
    • (a) Opposite angles of a cyclic quadrilateral add up to 180 degrees — Q1
    • (b) \(180 - 78\) — M1
    • (b) \(102\) — A1
  3. 3 Work out [3 marks]

    \(ABCD\) is a cyclic quadrilateral. Angle \(A = 4x - 5\) and angle \(C = 2x + 35\). Work out the value of \(x\). [3 marks]

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    Model answer

    Opposite angles add up to \(180^\circ\), so \(4x - 5 + 2x + 35 = 180\). Then \(6x = 150\) and \(x = 25\).

    Mark scheme

    • \((4x - 5) + (2x + 35) = 180\) — M1
    • \(6x + 30 = 180\) or \(6x = 150\) — M1
    • \(25\) — A1
  4. 4 Work out [3 marks]

    Not drawn accurately. \(ABCD\) is a cyclic quadrilateral. The side \(AB\) is extended to the point \(E\). Angle \(CBE = 60^\circ\). (a) Work out the size of angle \(ABC\). [1 mark] (b) Work out the size of angle \(ADC\). Give a reason for your answer. [2 marks]

    A circle diagram showing a cyclic quadrilateral with a side extended.
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    Model answer

    (a) \(ABC = 180 - 60 = 120^\circ\), because angles on a straight line add up to \(180^\circ\). (b) \(ADC = 180 - 120 = 60^\circ\), because opposite angles of a cyclic quadrilateral add up to \(180^\circ\).

    Mark scheme

    • (a) \(120\) — B1
    • (b) \(60\) — B1
    • (b) Opposite angles of a cyclic quadrilateral add up to 180 degrees — Q1
  5. 5 Work out [4 marks]

    \(A\), \(B\), \(C\) and \(D\) are points on the circumference of a circle. The lines \(AC\) and \(BD\) cross at \(E\). Angle \(CAD = 29^\circ\) and angle \(ABC = 97^\circ\). (a) Work out the size of angle \(CBD\). Give a reason for your answer. [2 marks] (b) Work out the size of angle \(ADC\). Give a reason for your answer. [2 marks]

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    Model answer

    (a) \(CBD = 29^\circ\), because angles in the same segment are equal. (b) \(ADC = 180 - 97 = 83^\circ\), because opposite angles of a cyclic quadrilateral add up to \(180^\circ\).

    Mark scheme

    • (a) \(29\) — B1
    • (a) Angles in the same segment are equal — Q1
    • (b) \(83\) — B1
    • (b) Opposite angles of a cyclic quadrilateral add up to 180 degrees — Q1
  6. 6 Show that [3 marks]

    \(ABCD\) is a parallelogram. Angle \(A = 70^\circ\). Show that \(ABCD\) cannot be a cyclic quadrilateral. [3 marks]

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    Model answer

    Opposite angles of a parallelogram are equal, so angle \(C = 70^\circ\). Then \(70 + 70 = 140\), which is not \(180^\circ\), so the opposite angles do not add up to \(180^\circ\) and \(ABCD\) is not cyclic.

    Mark scheme

    • Opposite angles of a parallelogram are equal, so \(C = 70^\circ\) — M1
    • \(70 + 70 = 140\) — M1
    • States that 140 is not 180, so the quadrilateral is not cyclic — Q1

Quick check

  1. 1

    Two angles are in the same segment of a circle. One is \(47^\circ\). What is the other?

    1. A\(133^\circ\)
    2. B\(94^\circ\)
    3. C\(47^\circ\)
    4. D\(43^\circ\)
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    C: \(47^\circ\)

    Angles in the same segment are equal.

  2. 2

    What do opposite angles of a cyclic quadrilateral add up to?

    1. A\(360^\circ\)
    2. B\(180^\circ\)
    3. C\(90^\circ\)
    4. D\(270^\circ\)
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    B: \(180^\circ\)

    This is the cyclic quadrilateral theorem.

  3. 3

    A cyclic quadrilateral has an angle of \(112^\circ\). What is the opposite angle?

    1. A\(68^\circ\)
    2. B\(112^\circ\)
    3. C\(248^\circ\)
    4. D\(78^\circ\)
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    A: \(68^\circ\)

    \(180 - 112 = 68\).

  4. 4

    What is a cyclic quadrilateral?

    1. AA quadrilateral with all sides equal
    2. BA quadrilateral with a circle inside it
    3. CA quadrilateral with four right angles
    4. DA quadrilateral with all four corners on a circle
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    D: A quadrilateral with all four corners on a circle

    “Cyclic” means all the corners lie on one circle.

  5. 5

    In a cyclic quadrilateral \(ABCD\), \(\angle A = 2x + 10\) and \(\angle C = 3x + 20\). What is \(x\)?

    1. A\(10\)
    2. B\(15\)
    3. C\(30\)
    4. D\(50\)
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    C: \(30\)

    \(5x + 30 = 180\), so \(5x = 150\) and \(x = 30\).

  6. 6

    \(ABCD\) is cyclic and the side \(AB\) is extended to \(E\). Angle \(CBE = 70^\circ\). What is angle \(ADC\)?

    1. A\(110^\circ\)
    2. B\(70^\circ\)
    3. C\(35^\circ\)
    4. D\(140^\circ\)
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    B: \(70^\circ\)

    An exterior angle of a cyclic quadrilateral equals the interior opposite angle.

  7. 7

    Which of these must be true for the angles \(ACB\) and \(ADB\) to be equal?

    1. A\(C\) and \(D\) are on the same side of the chord \(AB\)
    2. B\(C\) and \(D\) are on opposite sides of \(AB\)
    3. C\(AB\) is a diameter
    4. D\(C\), \(D\) and \(O\) are in a line
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    A: \(C\) and \(D\) are on the same side of the chord \(AB\)

    Angles in the same segment are made on the same side of a chord.

  8. 8

    A quadrilateral has opposite angles of \(95^\circ\) and \(80^\circ\). Can it be cyclic?

    1. AYes, because they are both less than \(180^\circ\)
    2. BYes, because all quadrilaterals are cyclic
    3. CNo, because the angles must be equal
    4. DNo, because \(95 + 80 \ne 180\)
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    D: No, because \(95 + 80 \ne 180\)

    Opposite angles of a cyclic quadrilateral must add up to \(180^\circ\), and \(95 + 80 = 175\).

  9. 9

    \(A\), \(B\), \(C\) and \(D\) are on a circle, with \(AC\) and \(BD\) meeting at \(E\). Angle \(CAD = 36^\circ\). What is angle \(CBD\)?

    1. A\(72^\circ\)
    2. B\(144^\circ\)
    3. C\(36^\circ\)
    4. D\(54^\circ\)
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    C: \(36^\circ\)

    Angles \(CAD\) and \(CBD\) are made by the chord \(CD\) on the same side, so they are equal.