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Exam questions · Maths · Geometry and Measures

Angles in Parallel Lines and Polygons

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

    Three angles of a quadrilateral are \(85^\circ\), \(100^\circ\) and \(70^\circ\). Work out the fourth angle.

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    Model answer

    The angles in a quadrilateral add up to \(360^\circ\). \(85 + 100 + 70 = 255\), and \(360 - 255 = 105^\circ\).

    Mark scheme

    • \(85 + 100 + 70 = 255\) — M1
    • \(105^\circ\) — A1
  2. 2 Work out [3 marks]

    In triangle \(PQR\), angle \(P = 48^\circ\) and angle \(Q\) is twice the size of angle \(R\). Work out the size of angle \(Q\).

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    Model answer

    Angles \(Q\) and \(R\) add up to \(180 - 48 = 132^\circ\). Let angle \(R = r\), so \(3r = 132\) and \(r = 44^\circ\). Angle \(Q = 2 \times 44 = 88^\circ\).

    Mark scheme

    • \(180 - 48 = 132\) — M1
    • \(132 \div 3 = 44\) — M1
    • \(88^\circ\) — A1
  3. 3 Work out [4 marks]

    The diagram shows triangle \(ABC\). \(AB\) is extended to a point on a straight line. (a) Work out the size of angle \(x\). Give a reason for your answer. [3 marks] (b) Work out the size of angle \(ABC\). [1 mark]

    Triangle ABC with the side AB extended, showing an angle of 54 degrees at A, an exterior angle of 118 degrees at B and an unknown angle x at C.
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    Model answer

    (a) The exterior angle of a triangle equals the sum of the two interior opposite angles, so \(x = 118 - 54 = 64^\circ\). (b) Angles on a straight line add up to \(180^\circ\), so angle \(ABC = 180 - 118 = 62^\circ\).

    Mark scheme

    • (a) \(118 - 54\) — M1
    • (a) \(64^\circ\) — A1
    • (a) Exterior angle equals the sum of the two opposite interior angles — Q1
    • (b) \(62^\circ\) — B1
  4. 4 Work out [3 marks]

    Each interior angle of a regular polygon is \(150^\circ\). Work out the number of sides of the polygon.

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    Model answer

    The exterior angle is \(180 - 150 = 30^\circ\), and \(360 \div 30 = 12\) sides.

    Mark scheme

    • \(180 - 150 = 30\) — M1
    • \(360 \div 30\) — M1
    • 12 — A1
  5. 5 Work out [3 marks]

    A regular hexagon and a square are joined along a common side. The shapes do not overlap. Work out the size of the angle at one end of the common side that is outside both shapes.

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    Model answer

    The interior angle of a regular hexagon is \(120^\circ\) and of a square is \(90^\circ\). Angles around a point add up to \(360^\circ\), so the angle is \(360 - 120 - 90 = 150^\circ\).

    Mark scheme

    • \(120^\circ\) or \(90^\circ\) found — M1
    • \(360 - 120 - 90\) — M1
    • \(150^\circ\) — A1
  6. 6 Work out [3 marks]

    The interior angle of a regular polygon is 5 times the size of its exterior angle. Work out the number of sides of the polygon.

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    Model answer

    The interior and exterior angles add up to \(180^\circ\), so \(6 \times\) exterior \(= 180\) and the exterior angle is \(30^\circ\). Then \(360 \div 30 = 12\) sides.

    Mark scheme

    • Interior \(+\) exterior \(= 180\) — M1
    • Exterior angle \(= 30^\circ\) — M1
    • 12 — 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 \(80^\circ\).