Chapter 8 Shapes 8-1 Naming and measuring angles KEY IDEAS An angle is formed when two lines called arms meet at a point called the vertex. A U N SA C O M R PL R E EC PA T E G D ES arm B x arm vertex C When naming angles, three letters are usually used. The letters are positioned at the end points of the arms. This angle is named ‘angle ABC’ or ∠ABC, or AB̂C or ∠B or x. Note also that AB̂C is the same as CB̂A. Angles are measured using a protractor. The size is determined by measuring from the arm on 0o. This angle measures 72° not 108°. Angles are classified (grouped) according to their size. Angle type Angle size Acute angle An angle between 0° and 90° Right angle One quarter turn of 90° (indicated by a small square) Straight angle One half turn of 180° Obtuse angle An angle between 90° and 180° Reflex angle An angle between 180° and 360° Revolution or perigon One complete turn of 360° Example Chapter 8 75 Shapes Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 75 13/08/13 6:01 PM 1 Estimate and then measure each of the marked angles. Name each type of angle. b c U N SA C O M R PL R E EC PA T E G D ES a 2 76 Estimate Estimate Estimate Size Size Size Type Type Type Use a protractor to draw each of the following angles on the lines below. a 40° b 90° c 150° d 270° e 350° Mathematics & Statistics for the New Zealand Curriculum Workbook: Year 9 Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 76 13/08/13 6:01 PM Chapter 8 3 In this diagram measure ∠AOB, ∠BOC and ∠AOC. What do you notice about the sum Shapes A (addition) of the smaller two angles? C U N SA C O M R PL R E EC PA T E G D ES ∠AOB = ∠BOC = 4 B O ∠AOC = For a clock face with numbers 1 to 12, work out the following without using a protractor. a The angle the minute hand turns in: i 15 minutes iv 5 minutes b ii 1 hour v 55 minutes iii 20 minutes The angle between the hour hand and the minute hand at: i 6 p.m. iii 7 a.m. ii 9 a.m. iv 11 p.m. Chapter 8 77 Shapes Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 77 13/08/13 6:01 PM 8-2 Angles and lines KEY IDEAS Complementary and supplementary angles • Complementary angles sum to 90°. U N SA C O M R PL R E EC PA T E G D ES • Supplementary angles sum to 180°. a b x + y = 180° a + b = 90° Vertically opposite angles B A • Vertically opposite angles are on opposite sides of the vertex and are equal. • ∠AOB and ∠COD are vertically opposite angles and equal. • ∠AOC and ∠BOD are also vertically opposite angles and equal. C O Adjacent angles on a straight line are complementary. D B A This means that ∠AOC and ∠AOB sum to 180°. Angles around a point add to 360°. O This means that ∠AOB and ∠BOC and ∠COD and ∠DOA sum to 360°. C D A O B D C When a straight line crosses parallel lines three special angles are formed. • Alternate angles are equal. • Corresponding angles are equal. • Co-interior angles are supplementary. 78 Mathematics & Statistics for the New Zealand Curriculum Workbook: Year 9 Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 78 13/08/13 6:01 PM Chapter 8 1 Shapes Write S if the following groups of angles are supplementary, C if they are complementary and X of they are neither. 37°, 43° b 27°, 153° c 41°, 49° d 11°, 89° U N SA C O M R PL R E EC PA T E G D ES a 2 3 What is the size of the unknown angle x required to make these angles: i complementary? ii supplementary? a 54º, x i ii b 32º, x i ii c 51º, x i ii d x, 28º i ii For these diagrams, find the value of x without using a protractor. a b c x 152° 70° x 4 x 50° For these diagrams, find the value of x without using a protractor. a b c x 2x 2x 30° 3x 30° 5 54° For these diagrams find the value of x and y without using a protractor. a b 61° x y c x 42° y d 38° 112° y x x Chapter 8 y 79 Shapes Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 79 13/08/13 6:01 PM 8-3 Triangles KEY IDEAS Triangles can be described by considering side lengths and internal angles. U N SA C O M R PL R E EC PA T E G D ES Sides that have the same length are marked with similar dashes and equal angles are indicated with the same markings, as shown in this isosceles triangle. Triangles are classified by their side lengths. Shape Number of equal sides Number of equal angles Equilateral triangle 3 3 Isosceles triangle 2 2 Scalene triangle 0 0 Name Triangles can also be classified by their internal angles. Shape Name Angle properties Acute-angled triangle All angles acute Right-angled triangle One right angle Obtuse-angled triangle One obtuse angle The three angles inside a triangle always sum (add) to 180°. a b c a + b + c = 180° 1 Name these triangles as scalene, isosceles or equilateral. a 80 b c Mathematics & Statistics for the New Zealand Curriculum Workbook: Year 9 Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 80 13/08/13 6:01 PM Chapter 8 2 Shapes Describe these triangles by their angle properties: acute, right or obtuse-angled triangles. b c U N SA C O M R PL R E EC PA T E G D ES a 3 Calculate the size of all unknown angles in these triangles. Don’t use a protractor as they may not be drawn to scale. a b c x x 85° 20° 30° x 40° x 22° x d e f x y 35° y 140° x 4 y x For each of these triangles, write an equation using the given letters. x a x b y y 5 Use a protractor to measure the marked angles: a, b and c, and then calculate a + b. What do you notice? a c a= b= b c= a+b= Chapter 8 81 Shapes Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 81 13/08/13 6:01 PM 8-4 Quadrilaterals KEY IDEAS All four-sided two-dimensional shapes are called quadrilaterals. Quadrilaterals are described according to the properties of their sides and angles, in a similar way to the way in which triangles are described. U N SA C O M R PL R E EC PA T E G D ES Sides that are equal in length are shown with the same number of small dashes. Sides that are parallel are marked with the same number of arrow heads. Name Shape Special properties Parallelogram A parallelogram has two pairs of equal sides and two pairs of equal angles. Both pairs of opposite sides are parallel. Rhombus or diamond A rhombus or diamond is a special parallelogram with four sides equal, two pairs of equal angles and two pairs of parallel sides. Rectangle A rectangle has the same properties as a parallelogram but with each angle 90°. Square A square has the same properties as a rhombus but with each angle 90°. It also has the same properties as a rectangle but with all four sides equal. Trapezium A trapezium has one pair of opposite sides parallel. An isosceles trapezium will also have one pair of sides of equal length and two pairs of equal angles. Kite 82 A kite has two pairs of equal adjacent sides (adjacent means ‘beside’) and one pair of opposite angles equal. Mathematics & Statistics for the New Zealand Curriculum Workbook: Year 9 Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 82 13/08/13 6:01 PM Chapter 8 Shapes The four angles of a quadrilateral always sum (add) up to 360°. x y U N SA C O M R PL R E EC PA T E G D ES z w x + y + z + w = 360° 1 2 List all the quadrilaterals that have: a 2 pairs of parallel sides b 2 different pairs of equal length sides c only one pair of parallel sides d 4 right angles e 1 pair of opposite equal angles f 2 different pairs of equal opposite angles A rhombus becomes a square if all its angles are equal. What property needs to be added to the given shape if it is to be transformed (changed) into the shape listed in the brackets? 3 a Parallelogram (rectangle) b Rhombus (square) c Rectangle (square) d Trapezium (isosceles trapezium) e Kite (rhombus) Calculate the size of all unknown angles in these quadrilaterals. Don’t use a protractor as they may not be drawn to scale. a x b 100° x y 55° c y a b 120° 60° 40° 4 For this quadrilateral, write an equation using the given letters. x y Chapter 8 83 Shapes Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 83 13/08/13 6:01 PM 8-5 Other polygons KEY IDEAS Polygons are two-dimensional shapes. Polygons are described according to the number of their sides and whether they are equal. U N SA C O M R PL R E EC PA T E G D ES Sides that are equal in length are shown with the same number of small dashes. Polygons that have all their sides equal are called regular. Number of sides 3 4 5 6 7 8 10 12 Polygon name Triangle Quadrilateral Pentagon Hexagon Septagon Heptagon Octagon Decagon Dodecagon The exterior angles of a polygon sum to 360°. b a e c d a + b + c + d + e = 360° 1 84 Name these shapes. a b c d e f Mathematics & Statistics for the New Zealand Curriculum Workbook: Year 9 Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 84 13/08/13 6:01 PM Chapter 8 2 Shapes Calculate the marked angles. a b b 53° 48° U N SA C O M R PL R E EC PA T E G D ES a 88° 42° 61° c d 154° d c 92° 3 Complete this table. Name of regular shape Number of sides Size of each exterior angle Size of each interior angle 3 4 5 6 8 9 10 12 Chapter 8 85 Shapes Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 85 13/08/13 6:01 PM 8-6 Constructing shapes KEY IDEAS To construct a shape we can use either: • compasses or U N SA C O M R PL R E EC PA T E G D ES • protractors and rulers. 1 Construct and name the type of triangle in each case. a triangle ABC where AB = 3 cm, BC = 6 cm and CA = 8 cm 86 b triangle XYZ where XY = YZ = 5 cm, ZX = 3 cm 2 Using a compass and a ruler construct a right-angled triangle ABC with ∠B = 90°, BC = 4 cm and AB = 3 cm. 3 a Construct an equilateral triangle ABC with all sides 5 cm. b Construct an isosceles triangle ABC with AB = 2 cm, BC = 5 cm and AC = 5 cm. Mathematics & Statistics for the New Zealand Curriculum Workbook: Year 9 Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 86 13/08/13 6:01 PM Chapter 8 4 Shapes Draw these quadrilaterals using a ruler and protractor. b 3 cm a 3 cm 4 cm c U N SA C O M R PL R E EC PA T E G D ES 6 cm 5 cm 5 cm 70° 4 cm 5 Construct a hexagon with sides of 5 cm. Chapter 8 87 Shapes Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 87 13/08/13 6:01 PM 8-7 Three-dimensional shapes KEY IDEAS A prism is a three-dimensional shape with a constant (same) cross section. A net is the two-dimensional result of ‘unfolding’ a U N SA C O M R PL R E EC PA T E G D ES 3D shape. The plan view of an object is what it looks like from above. An isometric view of an object shows a 3D view of the object. Isometric paper has dots which are on lines at 60° angles. 1 2 Name these shapes and state whether they are prisms or not. a b c Name Name Name Prism yes/no Prism yes/no Prism yes/no d e f Name Name Name Prism yes/no Prism yes/no Prism yes/no Complete these nets so that when they are folded up to make dice, the opposite faces add to 7. a 1 2 3 b c 5 4 1 2 4 d 5 3 6 6 88 Mathematics & Statistics for the New Zealand Curriculum Workbook: Year 9 Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 88 13/08/13 6:01 PM Chapter 8 3 Shapes Draw the right, top and front view of these models. b c U N SA C O M R PL R E EC PA T E G D ES a 4 Draw these models on the isometric paper below. a b Chapter 8 89 Shapes Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400 9781107653573txt_03pp.indd 89 13/08/13 6:01 PM
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