15 Polygons Angle Facts. Example 1. Solution. Example 2. Solution
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1 15 Polygons MEP Y8 Practice Book B 15.1 Angle Facts In this section we revise some asic work with angles, and egin y using the three rules listed elow: The angles at a point add up to 360, e.g. a c a + + c = 360 The angles on a straight line add up to 180, e.g. e f e + f = 180 The angles in a triangle add up to 180, e.g. x w y w + x + y = 180 Example 1 Determine the size of angle a in the diagram shown a a = 360 (angle sum at a point) a = 360 a = 87 Example 2 Determine the size of angle d in the diagram shown d = 180 (angle sum in a triangle) 147 +d = 180 d = 33 d
2 Example 3 Determine the size of angle n in the diagram shown. n + 27 = 180 (angle sum on a line) n = 153 n 27 Exercises 1. Calculate the sizes of the angles marked y letters in the following diagrams: a (c) (d) c d 2. Calculate the sizes of the unknown angles in the following triangles: (c) (d)
3 15.1 MEP Y8 Practice Book B 3. Calculate the sizes of the angles marked y the letter x in the following diagrams: x x 81 (c) (d) x x The diagram shows an isosceles triangle. What are the sizes of the two angles marked a and? 40 a 5. Calculate the sizes of the angles marked a and in the diagram. 143 a The diagram opposite shows two intersecting straight lines. Calculate the sizes of the angles marked a, and c in the diagram. What do you notice aout angles a and c? a 47 c c 7. The diagram opposite shows a rectangle and its diagonals. Calculate the sizes of the angles marked a, and c. 37 a 37 54
4 8. Determine the sizes of the angles marked a, and c in the diagram shown. 72 a c 9. PQR is a straight line. Determine the sizes of the angles marked a, and c in the triangles shown. c a P Q R 10. Calculate the sizes of the angles marked a,, c, d and e in the triangles shown. c a d e 55
5 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 interior angles of the hexagon. are the The angles marked the exterior angles of the hexagon. are 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 aove 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. The exterior angles of a regular hexagon are all equal, as shown in the previous diagram. 56
6 Therefore the exterior angle of a regular hexagon = = 60 So the interior angle of a regular hexagon = The sum of the interior angles = Example 2 = 720 The exterior angle of a regular polygon is 40. Calculate: the size of the interior angle, the numer of sides of the polygon. Interior angle + exterior angle = 180 Interior angle = = 140 = 120 The numer of sides can e determined y dividing 360 y the size of the exterior angles, giving 360 = 9 40 so the polygon has 9 sides. In a regular polygon: exterior angle = numer of sides = 360 the numer of sides 360 exterior angle 57
7 15.2 MEP Y8 Practice Book B Exercises 1. Calculate the size of the exterior angles of a regular polygon which has interior angles of: (c) 162 (d) Calculate the sizes of the exterior and interior angles of: a regular octagon, a regular decagon. 3. Calculate the size of the interior angles of a regular 12-sided polygon. What is the sum of the interior angles of a regular 12-sided polygon? 4. What is the size of the interior angle of a regular 20-sided polygon? What is the sum of the interior angles of a regular 20-sided polygon? 5. Calculate the size of the exterior angle of a regular pentagon. 6. The size of the exterior angle of a regular polygon is 12. How many sides does this polygon have? 7. Calculate the numer of sides of a regular polygon with interior angles of: (i) 150 (ii) 175 (iii) 162 (iv) 174 Show why it is impossile for a regular polygon to have an interior angle of Complete the following tale for regular polygons. Note that many of the missing values can e found in the examples and earlier exercises for this unit. 58
8 Numer Exterior Interior Sum of Interior of Sides Angles Angles Angles (c) Descrie an alternative way to calculate the sum of the interior angles of a regular polygon. Draw and measure the angles in some irregular polygons. Which of the results in the tale are also true for irregular polygons? 9. The exterior angle of a regular polygon is 4. How many sides does the polygon have? What is the sum of the interior angles of the polygon? 10. A regular polygon has n sides. Explain why the exterior angles of the polygon are of size 360 n. Explain why the interior angles of the polygon are 180 (c) Write an expression for the sum of the interior angles n 59
9 15.3 Symmetry In this section we revise the symmetry of ojects and examine the symmetry of regular polygons. Example 1 Draw the lines of symmetry of each shape elow: The shape has 2 lines of The shape has 2 diagonal lines symmetry, one horizontal of symmetry, as shown elow: and the other vertical, as shown elow: Reminder The order of rotational symmetry is the numer of times in one rotation of 360 that a shape is identical to that of its starting position. Example 2 What is the order of rotational symmetry of each of the following shapes: (c) 60
10 The shape has rotational symmetry of order 1, meaning that it does not have rotational symmetry. (The shape cannot e rotated to another position within 360 and still look the same.) 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 aout 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 aout its centre. * * Example 3 A heptagon is a shape which has 7 sides. Draw a diagram to show the lines of symmetry of a regular heptagon. What is the order of rotational symmetry of a regular heptagon? A regular heptagon has 7 lines of symmetry, as shown in the following diagram:
11 15.3 MEP Y8 Practice Book B A regular heptagon has rotational symmetry of order 7. The order of rotational symmetry and the numer of lines of symmetry of any regular polygon is equal to the numer of sides. Exercises 1. 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. (c) (d) (e) (f) 62
12 2. State the order of rotational symmetry and the numer of lines of symmetry, for each of the following shapes: (c) (d) (e) (f) 3. Descrie fully the symmetries of the following shapes: 4. Descrie the symmetry properties of each of the following triangles: Equilateral Isosceles Scalene 63
13 15.3 MEP Y8 Practice Book B 5. How many lines of symmetry does a square have? Draw a diagram to show this information. What is the order of rotational symmetry of a square? 6. Copy and complete the following tale: Shape Order of Rotational Numer 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) What do you conclude from the information in the tale? 7. Draw a shape that has no lines of symmetry, ut has rotational symmetry of order Draw a shape that has at least one line of symmetry and no rotational symmetry. 9. Draw two regular polygons, one with an even numer of sides and one with an odd numer of sides. By drawing lines of symmetry on each diagram, show how the positions of the lines of symmetry differ etween odd- and even-sided regular polygons. 10. Draw an irregular polygon that has oth line and rotational symmetry. Show the lines of symmetry and the centre of rotation, and state its order of rotational symmetry. 64
14 15.4 Quadrilaterals There are many special types of quadrilaterals; the following tale lists some of them and their properties. Quadrilateral Rectangle Properties 4 right angles and opposite sides equal Square 4 right angles and 4 equal sides Parallelogram Two pairs of parallel sides and opposite sides equal Rhomus Parallelogram with 4 equal sides Trapezium Two sides are parallel Kite Two pairs of adjacent sides of the same length Example 1 List the quadrilaterals that have four sides all of the same length. Square and rhomus. Example 2 List the quadrilaterals that do not have two pairs of parallel sides. Kite and trapezium. Example 3 Which quadrilaterals have diagonals that are perpendicular to one another? The square, rhomus and kite have diagonals that cross at right angles. 65
15 15.4 MEP Y8 Practice Book B Exercises 1. Which quadrilaterals have diagonals that are the same length? 2. Which quadrilaterals have exactly two lines of symmetry? Draw diagrams to show these lines of symmetry. 3. Which quadrilaterals have rotational symmetry of order 2? 4. Which quadrilaterals can have exactly one line of symmetry? Draw diagrams to show them and the line of symmetry. 5. Name each of the following quadrilaterals: (c) (d) (e) (f) 66
16 6. Which quadrilaterals have diagonals that are not equal in length? 7. A quadrilateral has four sides of the same length. Copy and complete the following sentences: The quadrilateral must e a.... The quadrilateral could e a... if Which quadrilaterals have more than one line of symmetry? 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. 9. The following flow chart is used to identify quadrilaterals: START DOES IT HAVE 4 RIGHT ANGLES? ARE ALL THE SIDES THE SAME LENGTH? ARE OPPOSITE ANGLES EQUAL? ARE ALL THE SIDES THE SAME LENGTH? ARE 2 SIDES PARALLEL? ARE THE DIAGONALS PERPEND- ICULAR? A B C D E F G Which type of quadrilateral arrives at each of the outputs, A to G? 67
17 15.4 MEP Y8 Practice Book B 10. The following flow chart can e used to classify quadrilaterals, ut some question oxes are empty. Copy and complete the flow chart. START ARE THE DIAGONALS THE SAME LENGTH? ARE TWO SIDES PARALLEL? KITE RHOMBUS SQUARE RECTANGLE PARALLELOGRAM TRAPEZIUM OTHER 68
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