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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 Work out [2 marks]

    Three angles at a point are \(130^\circ\), \(95^\circ\) and \(x^\circ\). Work out the value of \(x\). Give a reason for your answer.

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    Model answer

    Angles around a point add up to \(360^\circ\), so \(x = 360 - 130 - 95 = 135\).

    Mark scheme

    • \(360 - 130 - 95\) or \(130 + 95 = 225\) — M1
    • \(x = 135\) with the reason angles around a point add up to \(360^\circ\) — A1
  2. 2 Work out [3 marks]

    Triangle \(ABC\) is isosceles with \(AB = AC\). Angle \(BAC = 40^\circ\). Work out the size of angle \(ABC\). Give a reason for each step of your working.

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    Model answer

    The angles in a triangle add up to \(180^\circ\), so angles \(B\) and \(C\) add up to \(180 - 40 = 140^\circ\). The base angles of an isosceles triangle are equal, so angle \(ABC = 140 \div 2 = 70^\circ\).

    Mark scheme

    • \(180 - 40 = 140\) — M1
    • \(140 \div 2 = 70\) — A1
    • Reasons given: angles in a triangle add up to 180 degrees and base angles of an isosceles triangle are equal — C1
  3. 3 Work out [4 marks]

    The diagram shows two parallel lines crossed by a straight line. (a) Write down the size of angle \(x\). Give a reason for your answer. (2 marks) (b) Work out the size of angle \(y\). (2 marks)

    Two parallel lines crossed by a transversal, with one angle of 62 degrees and two unknown angles x and y marked at the other crossing.
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    Model answer

    (a) \(x = 62^\circ\) because alternate angles are equal. (b) \(x\) and \(y\) lie on a straight line, so \(y = 180 - 62 = 118^\circ\).

    Mark scheme

    • (a) \(62^\circ\) — B1
    • (a) Alternate angles are equal — C1
    • (b) \(180 - 62\) or \(180 - x\) — M1
    • (b) \(118^\circ\) — A1
  4. 4 Work out [3 marks]

    Work out the size of one interior angle of a regular hexagon. You must show your working.

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    Model answer

    The exterior angle is \(360 \div 6 = 60^\circ\), so the interior angle is \(180 - 60 = 120^\circ\). Alternatively the sum of the angles is \((6 - 2) \times 180 = 720^\circ\) and \(720 \div 6 = 120^\circ\).

    Mark scheme

    • \(360 \div 6 = 60\) or \((6 - 2) \times 180 = 720\) — M1
    • \(180 - 60\) or \(720 \div 6\) — M1
    • \(120^\circ\) — A1
  5. 5 Work out [3 marks]

    Each exterior angle of a regular polygon is \(20^\circ\). (a) Work out the number of sides of the polygon. (2 marks) (b) Work out the size of one interior angle. (1 mark)

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    Model answer

    (a) \(360 \div 20 = 18\) sides. (b) \(180 - 20 = 160^\circ\).

    Mark scheme

    • (a) \(360 \div 20\) — M1
    • (a) 18 — A1
    • (b) \(160^\circ\) — B1
  6. 6 Work out [4 marks]

    The angles of a quadrilateral are \(x^\circ\), \(2x^\circ\), \((3x - 10)^\circ\) and \(4x^\circ\). Work out the size of the largest angle of the quadrilateral.

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    Model answer

    The angles add up to \(360^\circ\), so \(x + 2x + 3x - 10 + 4x = 360\). That gives \(10x - 10 = 360\), so \(x = 37\). The largest angle is \(4x = 148^\circ\).

    Mark scheme

    • \(x + 2x + 3x - 10 + 4x = 360\) — M1
    • \(10x = 370\) or \(10x - 10 = 360\) — M1
    • \(x = 37\) — A1
    • \(148^\circ\) — 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 \(80^\circ\).