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Angles in Polygons

discia.org 10 questions · 34 marks
Name Date
  1. Either paper 3 marks

    Here is a quadrilateral.

    x90°70°90°Diagram NOT accurately drawn

    Work out the size of the angle marked \(x\).

    Show answer
    \(110^\circ\)
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  2. Either paper 4 marks

    The diagram shows a hexagon.

    x130°115°110°110°120°Diagram NOT accurately drawn
    (a)

    Find the sum of the interior angles of a hexagon.

    (b)

    Find the size of the angle marked \(x\).

    Show answer
    (a) \(720^\circ\)
    (b) \(135^\circ\)
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  3. Either paper 3 marks

    Here is a pentagon.

    x98°108°133°108°Diagram NOT accurately drawn

    Find the size of the angle marked \(x\).

    Show answer
    \(93^\circ\)
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  4. Either paper 3 marks

    A regular polygon has \(40\) sides.

    (a)

    Work out the size of each exterior angle.

    (b)

    Work out the size of each interior angle.

    Show answer
    (a) \(9^\circ\)
    (b) \(171^\circ\)
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  5. Either paper 4 marks

    Here is a regular dodecagon.

    x
    (a)

    Calculate the size of one exterior angle of the dodecagon.

    (b)

    Calculate the size of the angle marked \(x\).

    Show answer
    (a) \(30^\circ\)
    (b) \(150^\circ\)
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  6. Either paper 2 marks

    The sum of the angles in any quadrilateral is \(360^\circ\).

    A quadrilateral can be split into 2 triangles, so \(2 \times 180^\circ = 360^\circ\).

    Idris writes \(3 \times 180^\circ = 540^\circ\), so the sum of the angles in any pentagon must be \(540^\circ\).

    Is Idris correct?

    Tick a box. Yes / No

    Show working to support your answer.

    Show answer
    Yes. The angles in a pentagon add up to \(540^\circ\), and Idris claimed \(540^\circ\).
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  7. Either paper 3 marks

    Part of a regular polygon is shown. The jagged edge is where the rest of it has been torn off.

    The diagram is not drawn accurately.

    x
    (a)

    Assume that the polygon is a nonagon.

    Calculate the size of angle \(x\).

    (b)

    In fact, the polygon has fewer sides than a nonagon.

    What does this mean about the size of angle \(x\)? Tick one box.

    A. It is the same as the answer to part (a)

    B. It is less than the answer to part (a)

    C. It is more than the answer to part (a)

    D. It could be any of the above

    Show answer
    (a) \(40^\circ\)
    (b) C. It is more than the answer to part (a)
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  8. Either paper 3 marks

    Each interior angle of a regular polygon is \(144^\circ\).

    Calculate the number of sides the polygon has.

    Show answer
    \(10\ \text{sides}\)
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  9. Either paper 4 marks

    The interior angle of a regular polygon with \(5\) sides is \(108^\circ\).

    Here is the regular polygon.

    (a)

    At each vertex of the polygon there is a reflex angle outside the polygon.

    Calculate the sum of the reflex angles at \(4\) of the vertices.

    (b)

    Harry is asked how many lines of symmetry a heptagon has. He assumes it is regular.

    State the number of lines of symmetry of a regular heptagon.

    (c)

    Omar then says, "What if the heptagon is not regular?"

    For a heptagon that is not regular, which statement is true? Write down the letter of the true statement.

    A. There could be 0 or 1

    B. There could be 0, 1 or 2

    C. There must be 0

    D. There could be any number up to 7

    Show answer
    (a) \(1008^\circ\)
    (b) \(7\ \text{lines}\)
    (c) A. There could be 0 or 1
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  10. Either paper 5 marks

    In the diagram, \(CD\) and \(FG\) are parallel lines. \(FC\) and \(GD\) are straight lines.

    47°133°FCGD

    The lines \(FC\) and \(GD\) are extended to meet at the point \(A\).

    Show that triangle \(ACD\) is isosceles.

    Give a reason for each stage of your working.

    Show answer
    \(\angle ADC = 180 - 133 = 47^\circ\) (angles on a straight line add up to \(180^\circ\)); \(\angle ACD = 47^\circ\) (corresponding angles, parallel lines); two equal angles, so the triangle is isosceles
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discia.org Angles in Polygons · sheet 1
Answers Angles in Polygons · sheet 1
  1. \(110^\circ\)
  2. (a) \(720^\circ\)
    (b) \(135^\circ\)
  3. \(93^\circ\)
  4. (a) \(9^\circ\)
    (b) \(171^\circ\)
  5. (a) \(30^\circ\)
    (b) \(150^\circ\)
  6. Yes. The angles in a pentagon add up to \(540^\circ\), and Idris claimed \(540^\circ\).
  7. (a) \(40^\circ\)
    (b) C. It is more than the answer to part (a)
  8. \(10\ \text{sides}\)
  9. (a) \(1008^\circ\)
    (b) \(7\ \text{lines}\)
    (c) A. There could be 0 or 1
  10. \(\angle ADC = 180 - 133 = 47^\circ\) (angles on a straight line add up to \(180^\circ\)); \(\angle ACD = 47^\circ\) (corresponding angles, parallel lines); two equal angles, so the triangle is isosceles
discia.org Angles in Polygons · sheet 1