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Exam questions · Maths · Geometry and Measures

Angles in Parallel Lines and Polygons

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

    Two straight lines cross. One of the angles is \(38^\circ\). (a) Write down the size of the angle directly opposite it. [1 mark] (b) Calculate the size of an angle next to it. [1 mark]

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    Model answer

    (a) Vertically opposite angles are equal, so \(38^\circ\). (b) Angles on a straight line add up to \(180^\circ\), so \(180 - 38 = 142^\circ\).

    Mark scheme

    • (a) \(38^\circ\) — B1
    • (b) \(142^\circ\) — B1
  2. 2 Calculate [4 marks]

    \(ABCDE\) is a regular pentagon. (a) Calculate the size of one interior angle of the pentagon. [2 marks] (b) Calculate the size of angle \(x\). [2 marks]

    A regular pentagon ABCDE with the diagonal AC drawn and the angle ACD marked x.
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    Model answer

    (a) The exterior angle is \(360 \div 5 = 72^\circ\), so the interior angle is \(180 - 72 = 108^\circ\). (b) Triangle \(ABC\) is isosceles, so angle \(BCA = (180 - 108) \div 2 = 36^\circ\). Then \(x = 108 - 36 = 72^\circ\).

    Mark scheme

    • (a) \(360 \div 5\) or \(540 \div 5\) — M1
    • (a) \(108^\circ\) — A1
    • (b) \((180 - 108) \div 2 = 36\) — M1
    • (b) \(72^\circ\) — A1
  3. 3 Calculate [3 marks]

    A regular polygon has 15 sides. (a) Calculate the size of one exterior angle. [1 mark] (b) Calculate the size of one interior angle. [2 marks]

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    Model answer

    (a) \(360 \div 15 = 24^\circ\). (b) \(180 - 24 = 156^\circ\).

    Mark scheme

    • (a) \(24^\circ\) — B1
    • (b) \(180 - 24\) — M1
    • (b) \(156^\circ\) — A1
  4. 4 Calculate [3 marks]

    \(ABCD\) is a parallelogram. Angle \(DAB = 70^\circ\). Calculate the size of angle \(ABC\). Give a reason for your answer.

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    Model answer

    Angles \(DAB\) and \(ABC\) are co-interior angles between the parallel sides \(AD\) and \(BC\), so they add up to \(180^\circ\). Angle \(ABC = 180 - 70 = 110^\circ\).

    Mark scheme

    • \(180 - 70\) — M1
    • \(110^\circ\) — A1
    • Reason: co-interior angles add up to 180 degrees — B1
  5. 5 Calculate [3 marks]

    The interior angles of a polygon add up to \(1260^\circ\). Calculate the number of sides of the polygon.

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    Model answer

    \((n - 2) \times 180 = 1260\), so \(n - 2 = 7\) and \(n = 9\).

    Mark scheme

    • \(1260 \div 180 = 7\) — M1
    • \(7 + 2\) — M1
    • 9 — A1
  6. 6 Calculate [4 marks]

    The angles of a pentagon are \(x^\circ\), \((x + 10)^\circ\), \(2x^\circ\), \((2x + 20)^\circ\) and \((3x - 30)^\circ\). Calculate the size of the largest angle of the pentagon.

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    Model answer

    The interior angles of a pentagon add up to \(540^\circ\), so \(x + x + 10 + 2x + 2x + 20 + 3x - 30 = 540\). That gives \(9x = 540\), so \(x = 60\). The angles are \(60^\circ\), \(70^\circ\), \(120^\circ\), \(140^\circ\) and \(150^\circ\), so the largest is \(150^\circ\).

    Mark scheme

    • \(540\) used as the total — M1
    • \(9x = 540\) — M1
    • \(x = 60\) — A1
    • \(150^\circ\) — A1

Quick check

  1. 1

    Three angles on a straight line are \(47^\circ\), \(68^\circ\) and \(x\). What is \(x\)?

    1. A\(115^\circ\)
    2. B\(65^\circ\)
    3. C\(75^\circ\)
    4. D\(245^\circ\)
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    B: \(65^\circ\)

    Angles on a straight line add up to \(180^\circ\). \(180 - 47 - 68 = 65\).

  2. 2

    Three angles of a quadrilateral are \(80^\circ\), \(95^\circ\) and \(110^\circ\). What is the fourth angle?

    1. A\(75^\circ\)
    2. B\(85^\circ\)
    3. C\(105^\circ\)
    4. D\(285^\circ\)
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    A: \(75^\circ\)

    The angles of a quadrilateral add up to \(360^\circ\). \(80 + 95 + 110 = 285\) and \(360 - 285 = 75\).

  3. 3

    Which type of angles are equal and form a Z shape between parallel lines?

    1. ACorresponding angles
    2. BCo-interior angles
    3. CVertically opposite angles
    4. DAlternate angles
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    D: Alternate angles

    Alternate angles are on opposite sides of the transversal, and make a Z shape.

  4. 4

    Two co-interior angles lie between parallel lines. One is \(72^\circ\). What is the other?

    1. A\(72^\circ\)
    2. B\(18^\circ\)
    3. C\(108^\circ\)
    4. D\(118^\circ\)
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    C: \(108^\circ\)

    Co-interior angles add up to \(180^\circ\), so \(180 - 72 = 108\).

  5. 5

    What is the sum of the interior angles of a hexagon?

    1. A\(540^\circ\)
    2. B\(720^\circ\)
    3. C\(900^\circ\)
    4. D\(1080^\circ\)
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    B: \(720^\circ\)

    A hexagon has 6 sides, so the sum is \((6 - 2) \times 180 = 720\).

  6. 6

    What is each exterior angle of a regular octagon?

    1. A\(45^\circ\)
    2. B\(135^\circ\)
    3. C\(60^\circ\)
    4. D\(40^\circ\)
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    A: \(45^\circ\)

    \(360 \div 8 = 45\). The interior angle is \(135^\circ\).

  7. 7

    Each exterior angle of a regular polygon is \(24^\circ\). How many sides does it have?

    1. A14
    2. B16
    3. C156
    4. D15
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    D: 15

    \(360 \div 24 = 15\). The number 156 is the interior angle.

  8. 8

    An exterior angle of a triangle is \(118^\circ\). One of the interior opposite angles is \(54^\circ\). What is the other interior opposite angle?

    1. A\(62^\circ\)
    2. B\(72^\circ\)
    3. C\(64^\circ\)
    4. D\(172^\circ\)
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    C: \(64^\circ\)

    The exterior angle equals the sum of the two interior opposite angles, so \(118 - 54 = 64\).

  9. 9

    The angles of a triangle are \(x\), \(2x + 10\) and \(3x - 10\) degrees. What is the size of the largest angle?

    1. A\(30^\circ\)
    2. B\(80^\circ\)
    3. C\(70^\circ\)
    4. D\(90^\circ\)
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    B: \(80^\circ\)

    \(6x = 180\), so \(x = 30\). The angles are \(30^\circ\), \(70^\circ\) and \(80^\circ\).