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15 Polygons
15.1 Angle Facts
In this section we revise some basic work with angles , and begin by using the three rules listed below:
The angles at a point add up to 360 °, e.g.
a + b + c = 360 °
The angles on a straight line add up to 180 °, e.g.
e + f = 180 °
The angles in a triangle add up to 180 °, e.g.
w + x + y = 180 °
Example 1
Determine the size of angle a in the diagram shown.
Solution
81 ° + 92 ° + 100 ° + a = 360 ° (angle sum at a point)
a + 273 ° = 360 °
a = 87 °
Example 2
Determine the size of angle d in the diagram shown.
Solution
105 ° + 42 ° + d = 180 ° (angle sum in a triangle)
147 ° + d = 180 °
d = 33 °
a^ b c
e f
w y
x
81˚ 92˚ a 100˚
105˚
d 42˚
Example 3
Determine the size of angle n in the diagram shown.
Solution
n + 27 ° = 180 ° (angle sum on a line)
n = 153 °
Exercises
- Calculate the sizes of the angles marked by letters in the following diagrams:
(a) (b)
(c) (d)
- Calculate the sizes of the unknown angles in the following triangles:
(a) (b)
(c) (d)
27˚
n
123˚^ 110˚ a
71˚ b 89˚
37˚
33˚
87˚ d
77˚
113˚ 93˚ (^) 19˚ c 107˚
51˚
68˚
92˚
39˚ 43˚
33˚
111˚
- Determine the sizes of the angles marked a, b and c in the diagram shown.
- PQR is a straight line. Determine the sizes of the angles marked a, b and c in the triangles shown.
- Calculate the sizes of the angles marked a, b, c, d and e in the triangles shown.
72˚
19˚ 23˚
a b
c
P Q R
39˚
91˚ 61˚ (^) 85˚
c
a b
123˚
37˚
13˚ e
d
c
b
a
15.2 Angle Properties of Polygons
In this section we calculate the size of the interior and exterior angles for different regular polygons. The following diagram shows a regular hexagon:
The angles marked are the interior angles of the hexagon.
The angles marked are the exterior angles of the hexagon.
In a regular polygon the sides are all the same length and the interior angles are all the same size.
Note that, for any polygon: interior angle + exterior angle = 180 °.
Since the interior angles of a regular polygon are all the same size, it follows that the exterior angles are also equal to one another. One complete turn of the hexagon above will rotate any one exterior angle to each of the others in turn, which illustrates the following result:
The exterior angles of any polygon add up to 360 °.
Example 1
Calculate the sizes of the interior and the exterior angles of a regular hexagon. Hence determine the sum of the interior angles.
Solution
The exterior angles of a regular hexagon are all equal, as shown in the previous diagram.
Exercises
- Calculate the size of the exterior angles of a regular polygon which has interior angles of:
(a) 150 °
(b) 175 °
(c) 162 °
(d) 174 °
- Calculate the sizes of the exterior and interior angles of: (a) a regular octagon, (b) a regular decagon.
- (a) Calculate the size of the interior angles of a regular 12-sided polygon. (b) What is the sum of the interior angles of a regular 12-sided polygon?
- (a) What is the size of the interior angle of a regular 20-sided polygon? (b) What is the sum of the interior angles of a regular 20-sided polygon?
- Calculate the size of the exterior angle of a regular pentagon.
- The size of the exterior angle of a regular polygon is 12 °. How many sides does this polygon have?
- Calculate the number of sides of a regular polygon with interior angles of:
(a) (i) 150 ° (ii) 175 °
(iii) 162 ° (iv) 174 °
(b) Show why it is impossible for a regular polygon to have an interior angle of 123 °.
- (a) Complete the following table for regular polygons. Note that many of the missing values can be found in the examples and earlier exercises for this unit.
(b) Describe an alternative way to calculate the sum of the interior angles of a regular polygon. (c) Draw and measure the angles in some irregular polygons. Which of the results in the table are also true for irregular polygons?
- The exterior angle of a regular polygon is 4 °.
(a) How many sides does the polygon have? (b) What is the sum of the interior angles of the polygon?
- A regular polygon has n sides.
(a) Explain why the exterior angles of the polygon are of size
n
(b) Explain why the interior angles of the polygon are 180
n
(c) Write an expression for the sum of the interior angles.
Number Exterior Interior Sum of Interior of Sides Angles Angles Angles
Solution
(a) The shape has rotational symmetry of order 1, meaning that it does not have rotational symmetry. (The shape cannot be rotated to another position within 360 ° and still look the same.)
(b) The shape has rotational symmetry of order 4.
The following diagram shows how the position of one corner, marked *, moves as the square is rotated anticlockwise about its centre.
(c) The shape has rotational symmetry of order 2. The diagram shows the position of a corner, marked *, as the shape is rotated about its centre.
Example 3
A heptagon is a shape which has 7 sides.
(a) Draw a diagram to show the lines of symmetry of a regular heptagon.
(b) What is the order of rotational symmetry of a regular heptagon?
Solution
(a) A regular heptagon has 7 lines of symmetry, as shown in the following diagram: 1 2 3 4 5 6 7
(b) A regular heptagon has rotational symmetry of order 7.
The order of rotational symmetry and the number of lines of symmetry of any regular polygon is equal to the number of sides.
Exercises
- Copy each of the following shapes and draw in all the lines of symmetry. For each one, state the order of rotational symmetry and mark on your copy its centre of rotation. (a) (b)
(c) (d)
(e) (f)
- (a) How many lines of symmetry does a square have? Draw a diagram to show this information. (b) What is the order of rotational symmetry of a square?
- (a) Copy and complete the following table:
(b) What do you conclude from the information in the table?
- Draw a shape that has no lines of symmetry, but has rotational symmetry of order 3.
- Draw a shape that has at least one line of symmetry and no rotational symmetry.
- Draw two regular polygons, one with an even number of sides and one with an odd number of sides. By drawing lines of symmetry on each diagram, show how the positions of the lines of symmetry differ between odd- and even-sided regular polygons.
- Draw an irregular polygon that has both line and rotational symmetry. Show the lines of symmetry and the centre of rotation, and state its order of rotational symmetry.
Shape Order of Rotational Number of Lines Symmetry of Symmetry
Equilateral triangle
Square
Regular pentagon
Regular hexagon
Regular heptagon (7 sides)
Regular octagon
Regular nonagon (9 sides)
Regular decagon (10 sides)
Regular dodecagon (12 sides)
Quadrilateral Properties
Rectangle 4 right angles and opposite
Square 4 right angles and 4 equal sides
Parallelogram Two pairs of parallel sides and
Rhombus Parallelogram with 4 equal sides
Trapezium Two sides are parallel
Kite Two pairs of adjacent sides
15.4 Quadrilaterals
There are many special types of quadrilaterals; the following table lists some of them and their properties.
Example 1
List the quadrilaterals that have four sides all of the same length.
Solution
Square and rhombus.
Example 2
List the quadrilaterals that do not have two pairs of parallel sides.
Solution
Kite and trapezium.
Example 3
Which quadrilaterals have diagonals that are perpendicular to one another?
Solution
The square, rhombus and kite have diagonals that cross at right angles.
of the same length
opposite sides equal
sides equal
- Which quadrilaterals have diagonals that are not equal in length?
- A quadrilateral has four sides of the same length. Copy and complete the following sentences: (a) The quadrilateral must be a .....................
(b) The quadrilateral could be a .................... if .....................
- (a) Which quadrilaterals have more than one line of symmetry?
(b) Draw diagrams to show them and their lines of symmetry. (c) Which quadrilaterals have rotational symmetry of order greater than 1? List these quadrilaterals and state the order of their rotational symmetry.
- The following flow chart is used to identify quadrilaterals:
Which type of quadrilateral arrives at each of the outputs, A to G?
NO
START
NO
YES
YES
NO YES
YES
DOES IT HAVE 4 RIGHT ANGLES ?
ARE ALL THE SIDES THE SAME LENGTH ?
ARE 2 SIDES PARALLEL ? ARE ALL THE SIDES THE SAME LENGTH ?
ARE OPPOSITE ANGLES EQUAL ?
NO
NO
YES
NO
YES
A B C D E F G
ARE THE DIAGONALS PERPEND- ICULAR ?
- The following flow chart can be used to classify quadrilaterals, but some question boxes are empty. Copy and complete the flow chart.
KITE
NO
START
NO
YES
NO
YES
ARE THE DIAGONALS THE SAME LENGTH ?
RHOMBUS
YES
SQUARE PARALLELOGRAM
NO
RECTANGLE TRAPEZIUM
NO
NO
OTHER
YES
YES (^) ARE TWO SIDES PARALLEL ?
YES
