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6 Geometry and space Australian Curriculum links Year 6 Proficiency strands Problem-solving includes calculating angles Reasoning includes investigating new situations using known properties of angles Content descriptions Measurement and geometry Shape Elaborations Construct simple prisms and pyramids (ACMMG140) • considering the history and significance of pyramids from a range of cultural perspectives including those structures found in China, Korea and Indonesia • constructing prisms and pyramids from nets, and skeletal models Year 7 Content descriptions Measurement and Geometry Shape Elaborations Draw different views of prisms and solids formed from combinations of prisms (ACMMG161) • using aerial views of buildings and other 3-D structures to visualise the structure of the building or prism Geometric reasoning Elaborations Classify triangles according to their side and angle properties and describe quadrilaterals (ACMMG165) • identifying side and angle properties of scalene, isosceles, right-angled and obtuse angled triangles • describing squares, rectangles, rhombuses, parallelograms, kites and trapeziums Demonstrate that the angle sum of a triangle is 1808 and use this to find the angle sum of a quadrilateral (ACMMG166) • using concrete materials and digital technologies to investigate the angle sum of a triangle and quadrilateral Identify corresponding, alternate and cointerior angles when two parallel straight lines are crossed by a transversal (ACMMG163) • defining and classifying angles such as acute, right, obtuse, straight, reflex and revolution, and pairs of angles such as complementary, supplementary, adjacent and vertically opposite • constructing parallel and perpendicular lines using their properties, a pair of compasses and a ruler, and dynamic geometry software Investigate conditions for two lines to be parallel and solve simple numerical problems using reasoning (ACMMG164) • defining and identifying alternate, corresponding and allied angles and the relationships between them for a pair of parallel lines cut by a transversal, including using dynamic geometry software Year 8 Proficiency strands Reasoning includes using congruence to deduce properties of triangles Content descriptions Measurement and geometry Geometric reasoning Elaborations Define congruence of plane shapes using transformations (ACMMG200) • understanding the properties that determine congruence of triangles and recognising which transformations create congruent figures • establishing that two figures are congruent if one shape lies exactly on top of the other after one or more transformations (translation, reflection, rotation), and recognising the equivalence of corresponding sides and angles 06 MW7NC_TRB Final.indd 241 241 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Develop the conditions for congruence of triangles (ACMMG201) • constructing triangles using the conditions for congruence • solving problems using the properties of congruent figures, justifying reasoning and making generalisations • investigating the minimal conditions needed for the unique construction of triangles, leading to the establishment of the conditions for congruence (SSS, SAS, ASA and RHS), and demonstrating which conditions do not prescribe congruence (ASS, AAA) • plotting the vertices of two-dimensional shapes on the Cartesian plane, translating, rotating or reflecting the shape and using coordinates to describe the transformation Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning (ACMMG202) • establishing the properties of squares, rectangles, parallelograms, rhombuses, trapeziums and kites • identifying properties related to side lengths, parallelism, angles, diagonals and symmetry Source: Australian Curriculum Pre-test Most students should be able to recognise common geometric shapes by their appearance, even if they are not aware of geometric properties at this stage. Students need to be able to recognise shapes when they are represented in a non-standard position. For example, some students are unable to recognise a square as a square if it is tilted. Similarly, some students think shapes J and K in question 8 are triangles because they are ‘pointy’, rather than noticing that they each have more than three sides. 1 ● Look at this set of letters. a Which of the letters have perpendicular lines? b Which letters have parallel lines? c Which letters have both parallel and perpendicular lines? 2 ● 3 ● 4 ● What word could be used to describe this set of lines? Which of these things are usually vertical and which are usually horizontal? a tabletop b floor c classroom walls d ceiling e whiteboard Look again at the letters in question 1. a Which of the letters contain right angles? b In which of the letters can you see acute angles? c In which of the letters can you see obtuse angles? 242 06 MW7NC_TRB Final.indd 242 19/12/11 5:12 PM For each of these angles i name the angle using the given letters. ii what is the size of each of these angles? 40 40 30 40 0 90 80 7 0 10 10 01 2 01 80 90 100 110 13 12 60 01 30 50 0 170 180 10 20 K 160 30 50 0 170 180 10 20 160 30 R 70 60 40 30 50 50 J 50 8 ● 30 Q L Use your protractor to measure each of these angles and state whether the angle is an acute, obtuse or reflex angle. a 7 ● 60 01 01 13 12 40 0 10 2 0 100 90 80 70 110 14 0 2 01 80 90 100 110 01 180 170 1 60 1 50 1 70 60 14 40 50 6 ● b P 0 10 2 0 a 180 170 1 60 1 50 1 5 ● 6 chapter Geometry and space c b How many degrees are in each of the following? a a straight angle b one revolution about a point c a right angle. Which of the following shapes are a triangles? b quadrilaterals? A B C D E F G H I J K L 243 06 MW7NC_TRB Final.indd 243 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 9 ● 10 ● 11 ● Which of these shapes are squares? A B C D Consider the shapes on the right. a This triangle has three equal sides. What do we call this type of triangle? b This quadrilateral has both pairs of opposite sides equal and all the angles are right angles. What do we call this type of quadrilateral? Name these three-dimensional shapes. a b c 6 cm 6 cm 12 ● 6 cm Which of these shapes are prisms? A B C D Ans wer s 1 ● 2 ● 3 ● 4 ● 5 ● 6 ● 7 ● 8 ● a E, F, H, L, T b E, F, H, Z c E, F, H Parallel Horizontal: tabletop, floor, ceiling Vertical: classroom walls, whiteboard a E, F, H, L, T a i b i /PQR or /RQP /JKL or /LKJ b A, K, V, W, X, Y, Z c A, K, X, Y ii 80° ii 145° a 808, acute b 3048, reflex c 1058, obtuse a 1808 b 3608 c 908 a C, F, L b A, B, D, G, H, I, J 244 06 MW7NC_TRB Final.indd 244 19/12/11 5:12 PM 9 ● 10 ● 11 ● 12 ● 6 chapter Geometry and space A, D a equilateral triangle b rectangle a cube b square pyramid c hexagonal prism B, C Warm-up: parking problem Chapter warm-ups are included in the teacher and student ebooks as separate worksheet for ease of printing. This chapter warm-up links with the analysis task Parking problem at the end of the chapter, where students investigate the advantages and disadvantages of each type of parking. The parking signs below indicate whether parallel parking or angle parking applies. Parallel parking is where the cars park end-to-end parallel to the kerb (the edge of the roadway), as shown in the photograph, and angle parking is where the cars park at an angle to the kerb, usually 458, 608 or 908. a Find out the type of parking––angle parking or parallel parking––that applies in the road outside your school, in the shopping street of your town or suburb, or in the road where you live. b Why do you think this type of parking has been used? 245 06 MW7NC_TRB Final.indd 245 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Teaching note: the van Hiele theory of geometry learning When thinking about the teaching and learning of geometry, it is useful to gain an understanding of the levels of student understanding identified by Pierre van Hiele and Dina van Hiele-Geldof. In the 1950s this Dutch couple extensively explored children’s learning of geometry concepts. Russian mathematics educators found their work of interest, but it was not until the 1980s that the van Hiele theory came to the attention of American researchers. The theory has now been the subject of research world-wide and various aspects of the theory have been criticised. There has been a renumbering of the original levels, so that the van Hieles’ Level 0 is now Level 1. John Pegg has also suggested the splitting of the current Level 2 (old Level 1) into two levels. The van Hiele theory has also been criticised for its hierarchical nature of learning (that children do not progress to a higher level until they have mastered the previous level) and research has now shown that children can be at different levels for different geometry concepts. However, it does provide a useful framework for thinking about the development of geometric understanding. For further reading see Pegg, J (1995), ‘Learning and teaching geometry’, in: Grimison L & Pegg J, Teaching secondary school mathematics: theory into practice. Sydney, Harcourt Brace, 1995. In the currently used numbering, the van Hiele levels may be summarised as follows: Level Characteristic of the level Example Level 1 Recognising shapes by their visual appearance alone That shape is a square. Level 2 Level 2a Level 2b Recognising shapes by their properties Recognising one property Recognising more than one property Level 3 Recognising relationships between properties If all the angles of a square are right angles, then the opposite sides must be parallel. Level 4 Deductive reasoning and understanding the minimum properties that will identify a shape A rhombus with one right angle is sufficient to identify a shape as a square (why?); students are able to complete proofs such as the angle sum of a triangle. The shape is a square because it has four equal sides The shape is a square because it has four right angles, opposite sides are parallel, all sides are equal. Many students go through secondary school with a Level 1 understanding of geometry. The MathsWorld Australian Curriculum books place a strong emphasis on developing students’ thinking and understanding beyond Level 1. 246 06 MW7NC_TRB Final.indd 246 19/12/11 5:12 PM 6.1 Lines, rays and segments Teaching note: parallel lines Many students will be familiar with parallel bars in the playground or in gymnastics. It is important for students to recognise that the definition ‘Parallel lines are lines that will never meet even if they are extended in either direction’ is incomplete. This is the case only if the lines are in the same plane. We can also say that the perpendicular distance between parallel lines is the same at all points along the lines. Question 13 in exercise 6.1 is worth doing as a class activity, with each pair of students having a rectangular cardboard box, such as a shoe box or cornflakes packet. This provides an excellent means of conveying the notion that lines that never meet are not necessarily parallel. This idea can also be developed by looking at lines in the classroom, for example a vertical wall edge and its opposite horizontal ceiling-wall edge. However, the hands-on approach of looking at edges and drawing lines on a box helps students to see directly that there are line segments that will never meet, even if extended, that are not parallel. We must specify then that parallel lines are lines in the same plane that will never meet. A B D C F H G Introducing symbols Symbols for parallel and perpendicular are introduced in drawings as well as in written statements, for example AB || CD. Extra example 1 A four-sided shape is shown on the right. b a Which sides of this shape appear to be parallel? b Mark the diagram with symbols to show this. a c d Working Reasoning a Sides b and d appear to be parallel. These two sides b are the same distance apart at all positions. ■ are in the same plane. ■ do not meet. We use arrowheads to show that two lines or line segments are parallel. b a ■ c d 247 06 MW7NC_TRB Final.indd 247 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Extra example 2 The diagram on the right shows two line segments, AB and CD. C a What word can we use to describe AB and CD? B b Mark the diagram with a symbol to show this. 90° A D Working Reasoning a AB and CD are perpendicular. Lines are perpendicular if there is an angle of 908 between them. b The right angle symbol shows that the segments are perpendicular. C B 90° A D Teaching note: compass and ruler constructions Compasses are not always appropriate in classrooms. The Math Open Reference Project website has excellent animations of compass and ruler constructions that can be paused and replayed. Scrolling down the page leads to a clearly set out proof of why each construction works. www.mathopenref.com/tocs/constructionstoc.html The following are direct links to constructing a line perpendicular to a given line passing through a point on the line (as in example 3 in the student book) and passing through an external point (as in example 4 in the student book). www.mathopenref.com/constperplinepoint.html www.mathopenref.com/constperpextpoint.html exercise 6.1 Below are the answers to the questions in exercise 6.1 in the student book. Ans wer s 1 ● Line Ray Segment 248 06 MW7NC_TRB Final.indd 248 19/12/11 5:12 PM 2 ● 3 ● 4 ● 6 There are parallel segments in the rows of bricks. The legs of the parallel bars are parallel to each other. a CD, EF chapter Geometry and space 6.1 b AB, CD, EF A C B D 5 ● P R S Q 6 ● In striped material, the stripes are parallel to each other. In check material the horizontal stripes are perpendicular to the vertical stripes. 7 ● F A 8 ● 9 ● 10 ● C B D E Chris is building a house with rectangular rooms. He has checked that the floor is horizontal. The walls must be vertical and perpendicular to the floor. The walls on opposite sides of the room must be parallel to each other. Two walls meeting at a corner of the room must be perpendicular to each other. a perpendicular b vertical a c parallel d horizontal b c 11 ● a meet c don’t meet, but not parallel b don’t meet, but not parallel d don’t meet and parallel 249 06 MW7NC_TRB Final.indd 249 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 12 ● a Bay Street, Bank Street or Junction Street b Main Street, Tower Street or River Street 13 ● Banksia Street Railway Avenue Grant Street exercise 6.1 2 ● 3 ● Explain why you think this shape is called a parallelogram. These two lines are parallel. a Measure how far apart the lines are. b Show on the diagram where you measured to find how far apart the lines are. Jess is making a corduroy teddy bear for her little brother. She notices that each of the paper pattern pieces has a line labelled ‘Place on straight grain of fabric’. Using the word parallel, explain in a sentence what you think ‘Place on straight grain of fabric’ means. Paper pattern piece Place on straight grain of fabric 1 ● additional questions CUT 2 BODY FRONT Corduroy fabric 250 06 MW7NC_TRB Final.indd 250 19/12/11 5:12 PM Ans wer s 1 ● 2 ● 6 chapter Geometry and space 6.1 Both pairs of opposite sides of the shape are parallel. a 2 cm b 2 cm 3 ● Place the arrow on the pattern parallel to the direction of the lines in the fabric. 251 06 MW7NC_TRB Final.indd 251 19/12/11 5:12 PM 6.2 Angles Teaching note: dual concept of angle It is important that students understand the dual nature of the concept angle, ■ a dynamic concept of an amount of turning, for example as a door opens, ■ and the more familiar static concept of an angle at a vertex. Plastic geostrips or strips of card and paper fasteners are useful in developing the concept of an angle as an amount of turning. Plastic geostrips are available from Modern Teaching Aids, www.teaching.com.au. Another useful approach is to take students outside where there is room to spread out and ask them to make angles with their arms. If they start by putting both arms out horizontally together to the left, they can then move their right arm in a clockwise direction to make approximate representations of given angles. Teaching note: developing angle sense Students enjoy playing a game of ‘Simon Says’. Instruct them to close their eyes, turn clockwise through a right angle, turn anticlockwise through a straight angle, and so on. Students can vary this by making the angles with outstretched arms rather than turning their body. By concentrating so hard on the direction and type of angle, students make enough errors for this to be fun. Teaching note: symbol sense Students should be encouraged to use correct mathematical language. They will see that we use the same symbol for perpendicular and a right angle because perpendicular means at right angles to. Teaching note: estimating angles As well as being able to measure angles accurately, students should also be able to estimate angle sizes. A useful activity is for students to work in pairs, each carefully drawing a set of angles using pencil, ruler and protractor. Each student keeps a record of the sizes of the angles they have constructed. The angle sheets can then be swapped, with each student estimating the sizes of the angles drawn by the other student, then comparing their estimates with the constructed angle sizes. This can also be used as an exercise in accurate construction. Teaching note: measuring angles Three angle-measurers used by various trades people are shown on page 256 of the student book. There may be students in your class whose parents use one of these in their regular work. Students could think about situations where angles are important. The Year 5 Australian Curriculum includes ‘measuring and constructing angles using both 1808 and 3608 protractors’. Measuring angles with each protractor is included here for those students who do not already have this skill. 252 06 MW7NC_TRB Final.indd 252 19/12/11 5:12 PM 6 Students should think about the type of angle before they start, so they can check that they have used the correct scale on their protractors. The 3608 protractor is easier to use when working with reflex angles. chapter Geometry and space 6.2 Extra example 3 Use a 3608 protractor to measure the following angles. a b c Working Reasoning a Place the protractor with its centre at the vertex of the angle, with the zero mark on one arm of the angle. Measure around from zero in a clockwise direction on the outer scale. 0 40 35 03 3 03 30 40 33 03 20 50 60 100 90 80 70 10 01 13 0 30 1 22 0 02 14 10 01 50 200 160 190 170 50 60 1 170 1 180 14 23 0 30 4 02 Check: the angle is an acute angle so its measure must be between 08 and 908. 12 02 12 24 50 26 280 270 260 250 80 90 100 110 290 70 00 02 60 03 10 03 0 270 280 29 20 31 00 31 40 30 350 3 40 50 0 32 10 2 0 0 20 02 21 190 20 0 The angle measures 738. b 0 270 280 29 26 250 40 02 0 22 50 14 0 10 60 1 340 350 10 170 1 Place the protractor with its centre at the vertex of the angle, with the zero mark on one arm of the angle. Measure around from zero in an anticlockwise direction on the inner scale. 0 0 02 0 190 0 22 21 31 03 00 60 30 290 70 280 270 260 250 02 14 40 24 80 90 100 110 50 01 30 03 20 33 10 2 0 350 3 40 Check: the angle is an obtuse angle so its measure must be between 908 and 1808. 50 02 30 20 190 20 0 1 30 180 31 50 40 170 1 30 00 60 03 160 0 90 80 70 10 10 1 20 32 23 03 30 01 12 The angle measures 1248. continued 253 06 MW7NC_TRB Final.indd 253 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Extra example 3 continued Reasoning c Place the protractor with its centre at the vertex of the angle, with the zero mark on one arm of the angle. Measure around from zero in a clockwise direction on the outer scale because it is the reflex angle that is required. 40 350 340 30 10 2 0 3 0 0 20 80 90 100 110 70 12 50 70 60 60 40 10 110 20 01 13 270 260 2 50 2 Check: the angle is a reflex angle so its measure must be between 1808 and 3608. 0 01 23 30 22 02 01 50 10 200 160 60 1 50 14 0 20 170 1 190 170 23 40 350 3 40 3 30 3 10 20 280 290 00 03 31 260 270 280 290 250 30 40 03 2 0 90 80 0 1 30 14 3 3 20 50 0 Working 180 0 200 21 02 19 The angle measures 2438. Extra example 4 Use a 3608 protractor to draw the following angles. a 1958 b 988 Working c 378 Reasoning a 01 40 23 0 30 01 50 20 23 40 30 20 10 190 20 02 10 2 60 1 50 14 0 01 13 2 0 50 0 Rule a line segment for one arm of the angle. Place the protractor with its centre at the right-hand end of the line segment as shown. Place a pencil mark at 1958. Join the pencil mark to the right hand of the line segment to make the other arm of the angle. 180 170 1 32 170 190 200 160 10 350 3 40 3 30 32 0 0 270 260 25 90 28 02 2 00 03 31 02 22 10 2 0 0 3 40 35 0 03 0 33 60 14 40 50 80 90 100 110 70 12 260 270 280 290 250 30 40 03 0 90 80 70 02 10 10 10 6 01 195° continued 254 06 MW7NC_TRB Final.indd 254 19/12/11 5:12 PM 6 Extra example 4 continued 6.2 Working Reasoning b 50 60 1 31 0 50 170 1 60 350 10 20 340 30 15 01 40 1 30 00 90 80 7 0 30 10 20 90 3 100 110 40 0 190 200 10 02 22 31 03 00 50 60 30 290 70 280 270 260 250 80 90 14 40 5 01 20 10 2 0 3 0 350 3 40 3 30 3 02 24 19 0 60 17 180 0 200 2 24 Rule a line segment and place the centre of the protractor at the left-hand end of the segment, with the protractor line exactly along the segment. Measure around from 0 to 988 and place a small mark. Remove the protractor and join the mark to the left-hand end of the line segment to form the angle. 03 22 0 23 260 270 280 2 32 0 02 01 chapter Geometry and space 0 20 100 11 01 13 98° c 03 40 31 13 0 23 0 50 350 3 40 10 350 20 30 40 13 01 23 20 02 40 110 250 100 90 80 70 260 270 50 60 3 20 10 2 14 0 33 20 50 40 03 Rule a line segment and place the centre of the protractor at the right-hand end of the segment, with the protractor line exactly along the segment. Measure around from 0 to 378 and place a small mark. Remove the protractor and join the mark to the right-hand end of the line segment to form the angle. 02 60 1 190 20 170 1 180 0 190 170 200 160 10 33 03 02 20 02 22 30 20 0 270 260 25 90 28 2 00 01 10 2 0 60 14 40 50 80 90 100 110 70 1 0 00 280 29 03 31 37° 255 06 MW7NC_TRB Final.indd 255 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book exercise 6.2 BLM This exercise includes questions that reinforce the concept that we must be directly in front of an angle to measure it, and that angles can be important for safety (for example in wheel chair ramps) and in sport. A blackline master is included in the teacher and student ebooks providing a larger diagram for question 13. Question 13 Ans wer s Below are the answers to the questions in exercise 6.2 in the student book. 1 ● 2 ● 3 ● 4 ● 5 ● 6 ● 7 ● 8 ● 9 ● 10 ● 11 ● a c e g i /TOP (or /POT), acute, 508 /BUS (or /SUB), reflex, 2968 /HDG (or /GDH), obtuse, 1638 /MOP (or /POM), right, 908 /HFK (or /KFH), straight, 1808 b d f h /JCP (or /PCJ), obtuse, 1308 /PAT (or /TAP), reflex, 2708 /XYZ (or /ZYX), acute, 258 /KTP (or /PTK), obtuse, 1308 a obtuse angle e obtuse angle b acute angle f reflex angle c reflex angle g revolution d right angle h straight a line e reflex angle b perpendicular f ray c obtuse angle g acute angle d line segment h parallel a 668 e 758 b 3018 f 1538 c 1258 g 2908 d 2408 h 1058 a 708 f 958 b 2608 g 2058 a 338 b 408 c 1508 h 638 d 458 i 2858 e 1328 j 3458 a The photograph at the left, because it has been taken from a position more directly in front of the beam. b approximately 308 c 608 a A wheelchair ramp cannot be too steep otherwise the person will go down the slope too quickly and will not be able to stop. The ski slope needs to be steep enough for the skier to build up enough speed to keep going. The escalator needs to be steep enough so that it does not take up too much horizontal space, but not so steep that people could fall over on it. b i 38 ii 308 iii 308 c The photograph has been taken from a position side on to the escalator. If the angle is to be measured accurately, the photograph must be taken from a position directly in front of the escalator, at right angles to the tiled wall. a 408 b 538; yes, the angle is greater than 508 a a = 49, b = 50, c = 51, d = 48, e = 41, f = 39, g = 36, h = 39 b answers will vary a obtuse b acute 35° 70° 256 06 MW7NC_TRB Final.indd 256 19/12/11 5:12 PM c reflex d reflex 6 e acute 155° chapter Geometry and space 6.2 25° 80° f acute 270° g obtuse h acute 313° 138° 12 ● 13 ● Answers will match question 11. a b mid-on A 61° 283° B 264° 110° 138° 142° 153° 165° 194° 168° 180° 172° 175° 185° exercise 6.2 1 ● additional questions Match each of the terms listed on the left with their correct description from the list on the right. Line segment an angle of 908 Ray lines that are at right angles to each other Parallel lines an angle of 1808 Perpendicular lines an angle less than 908 Acute angle part of a line with a definite starting point but no finishing point Obtuse angle an angle greater than 908 but less than 1808 Right angle part of a line with a definite starting point and finishing point Straight angle an angle of 3608 Reflex angle two lines which are always the same distance apart Revolution an angle greater than 1808 but less than 3608 257 06 MW7NC_TRB Final.indd 257 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 2 ● In a b c d e the following diagram name a pair of parallel line segments. a pair of perpendicular lines. an acute angle. an obtuse angle. a right angle. E F G A D C B Ans wer s 1 ● 2 ● a b c d e Line segment part of a line with a definite starting point and finishing point Ray part of a line with a definite starting point but no finishing point Parallel lines two lines which are always the same distance apart Perpendicular lines lines that are at right angles to each other Acute Angle an angle less than 908 Obtuse Angle an angle greater than 908 but less than 1808 Right Angle an angle of 908 Straight Angle an angle of 1808 Reflex Angle an angle greater than 1808 but less than 3608 Revolution an angle of 3608 FD and GC Answers will vary. Two answers are: FD and BE; GC and BE. Answers will vary. Two answers are: /FGC; /DEF. one of /AGC or /AFD one of /EDF or /ECG 258 06 MW7NC_TRB Final.indd 258 19/12/11 5:12 PM 6.3 Calculating angle sizes Teaching note: using geometric language Students should appreciate how appropriate geometric language (for example, ‘vertically opposite angles’, ‘supplementary angles’) facilitates communication. They should be given the opportunity to practise this language. This is one of the benefits of students using interactive geometry software such as GeoGebra, particularly when they are working in pairs. Using the names of the various software tools (for example, ‘perpendicular line’, ‘parallel line’, ‘midpoint’) reinforces correct language. GeoGebra: complementary and supplementary angles Complementary and supplementary angles The GeoGebra (and HTML) file Complementary and supplementary angles in the student and teacher ebooks allows the size of angles to be changed to see pairs of angles that are complementary or supplementary. GeoGebra: vertically opposite angles Vertically opposite angles The interactive GeoGebra file, Vertically opposite angles, is included in the teacher and student ebooks in both GeoGebra and HTML formats. Students will notice that any two adjacent angles make a straight angle. The reasoning about why vertically opposite angles are equal is based on the reasoning ‘if a 1 b 5 180 and a 1 c 5 180, then b and c must be equal’. 259 06 MW7NC_TRB Final.indd 259 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Teaching note: when do we put a degrees symbol? Pronumerals are introduced to refer to unknown angles as an alternative to using angle names such as /ABC. It should be noted that in this chapter the pronumerals used for angle sizes stand for a number—the number of degrees in the angle. The pronumerals already have a degrees sign after them on the diagram, so the answers will be numeric only, without the degrees sign; for example, a 5 50, not a 5 50°. Teaching note: using algebra Students could be introduced to equations informally as shown in extra example 5 part a and example 13 part b in the student book. Alternatively, they could use a visual or arithmetic approach and show their working on a copy of the diagram. Extra example 5 a Find the value of a. b ABC is a straight line. Find the size of /ABD. D 78° a° 36° A B C Working Reasoning a a 1 36 = 90 a = 90 2 36 a = 54 The two adjacent angles add to 908. b /ABD 1 /DBC /ABD 1 788 /ABD /ABD = = = = 1808 1808 1808 2 788 1028 ABC is a straight line. /ABD and /DBC are supplementary angles. Worksheet: pool angles The worksheet Pool angles is included in the student and teacher ebooks. Pool angles This activity uses a pool table as its context. A diagram first shows how the ball bounces off the edge of the table at the same angle as it strikes the table. Students are then asked to use their ruler and protractor to draw the path of the ball after one, two, three and four bounces. Pool angles If you have ever played pool you will know the importance of angles. When a ball hits the edge, it bounces away at the same angle, as shown. The drawing below shows a pool table. A ball is hit in the direction shown. a Use your ruler and protractor to find where the ball is going to hit after its first bounce. Carefully draw the path of the ball. 260 06 MW7NC_TRB Final.indd 260 19/12/11 5:12 PM 6 b Using your ruler and protractor, carefully draw the path of the ball for its second, third and fourth bounces. (Assume that the ball continues to bounce so that it forms equal angles with the edges of the table each time.) chapter Geometry and space 6.3 55° Ans wer s a b 55° 55° exercise 6.3 Below are the answers to the questions in exercise 6.3 in the student book. Ans wer s 1 ● 2 ● 3 ● 4 ● 5 ● a 1458 b 238 c 908 d 588 e 1718 f 428 a 218 b 1098 c 728 d 1608 e 1158 f 1258 a 768 b 658 c 948 d 1008 e 588 f 408 a c e g i a = 38, b a = 61, b a = 82, b a = 70, b k 5 72 = = = = 38, c = 114 110, c = 110 59, c = 39, d = 59 39, c = 71, d = 70 Angle Complement b d f h d = 26, e = 116, f = 64 a = 48, b = 132, c = 42 a = 110, b = 70, c = 70 a = 36 Angle Complement 148 768 328 588 888 28 758 158 458 458 158 758 908 08 18 898 261 06 MW7NC_TRB Final.indd 261 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 6 ● 7 ● 8 ● 9 ● 10 ● 11 ● 12 ● 13 ● 14 ● 15 ● 16 ● 17 ● a a = 50 b b = 10 Angle Supplement c c = 32 208 908 908 238 1578 1758 58 918 898 788 1028 1008 808 18 1798 b b = 86 a 1078 a d g j e e = 64 f f = 24 d d = 45 e e = 65 f f = 90 Angle Supplement 1608 a a = 32 d d = 45 c c = 138 b ADB is a straight line, so the angles must add to 1808 e = 46 m = 23 a = 45, b = 45, c = 85 x = 285 a a = 60; b = 120 b e h k d = 70 h = 68 x = 66 x = 97 y = 107 w = 33 x = 40 x = 90 c f i l b c = 60; d = 120; e = 60; f = 120 a 308 c 608 e Queensberry Street b 308; same size as angle at A d 1208; supplementary angles f Elizabeth Street, Swanston Street a = 24 a /AOB or /AOC or straight angle /AOD b /AOC 808 a 368 b Bec and Sarah’s total angle = 1088, Emma’s total angle = 1448 a Blue 908; Green 1208; Yellow 728 b Red 788 c 72° 120° 78° 90° exercise 6.3 additional questions Teaching note In question 2, students may need help in reasoning that at 10 minutes past 5, the hour hand will have moved towards 6. They have already calculated in part c that the hour hand moves 0.58 per minute, so in 10 minutes, the hour hand will have moved through 10 3 0.5°, that is, 58. Hence, the required angle is the angle between 2 and 5 (that is, 908) plus 58, making an angle of 958 between the two hands. 262 06 MW7NC_TRB Final.indd 262 19/12/11 5:12 PM 1 ● 2 ● a Through how many degrees does the minute hand of a clock turn each minute? b What is the size of the obtuse angle between the hour hand and the minute hand of a clock at exactly 5 o’clock? c Through how many degrees does the hour hand turn each minute? d What is the angle between the hour hand and the minute hand at exactly 10 minutes past 5 o’clock? chapter 6 Geometry and space 6.3 11 12 1 12 2 9 3 8 4 7 6 5 Justify your reasoning for each of the following. For example, you could copy the diagram and label the sizes of any angles you calculate along the way to finding the required angle. a Calculate the size of ⬔MFX. b Find the value of x. M Y F 2x° N 3 ● x° 55° X 60° G When rays of light meet a flat shiny surface, such as a mirror, they are reflected at the same angle as they meet the mirror. For example, the diagram below shows how the rays of light are reflected if they meet the mirror at an angle of 758. Ray of light meeting the mirror 75° Reflected ray of light 75° Mirror A periscope is a device for seeing around corners. It consists of a bent tube with two mirrors arranged so that the rays of light are bent by 908 by each mirror and that the light rays end up parallel to their original direction. Copy the drawing and show how you would place the two mirrors so that the rays of light follow the path shown through the periscope. Light rays Eye 263 06 MW7NC_TRB Final.indd 263 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 4 ● Calculate the size of the angles between the blades of this wind generator. Ans wer s 1 ● 2 ● 3 ● a 68 b 1508 c 0.58 d 958 a 1458, /MFX = /NFY, as vertically opposite angles are equal. b x = 40, 2x 1 x 1 60 5 180° as supplementary angles. Mirror 1 Light rays Eye Mirror 2 45° 4 ● 120° 264 06 MW7NC_TRB Final.indd 264 19/12/11 5:12 PM 6.4 Angles and parallel lines GeoGebra: parallel lines Parallel lines A second interactive GeoGebra (and HTML) file Parallel lines is also included in the teacher and student ebooks. This time the lines AB and CD have been constructed so that they are parallel. Students can now confirm the angle relationships they observed in the Lines and transversals diagram when AB and CD were dragged until they appeared to be parallel. It is useful for students to see both interactive diagrams, as they then understand that any two lines may be cut by a transversal, but that the special angle relationships of equal alternate and corresponding angles and supplementary cointerior angles occur only when the lines are parallel. The converse is of course true, too: if alternate angles or corresponding angles are equal or if cointerior angles are supplementary, then the lines must be parallel. GeoGebra: lines and transversals Lines and transversals The interactive GeoGebra file Lines and transversals is included in the student and teacher ebooks in both GeoGebra and HTML formats. Two lines and a transversal can be dragged on the screen to observe angle relationships. Students will see that vertically opposite angles are always equal. In this interactive diagram, the lines AB and CD have not been constructed to be parallel. As the two lines are dragged until they appear parallel, students will notice that other angle relationships become apparent. 265 06 MW7NC_TRB Final.indd 265 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Class activity: angles and parallel lines Class activity Angles and parallel lines The class activity Angles and parallel lines includes four GeoGebra screen images of a pair of lines cut by a transversal. In the first two screens the lines are not parallel, but students can observe the relationship between vertically opposite angles. In the second pair of screens, the lines have been dragged until they are parallel. Students can now observe the additional angle relationships that occur for parallel lines cut by a transversal. Extra example 6 Find the values of the pronumerals. 74° a° b° Working Reasoning a = 74 (Alternate angles) The angles marked a8 and 748 are alternate angles. As the lines are parallel, the angles are equal. b = 106 (Corresponding angles) The angles marked 748 and b8 are corresponding angles. As the lines are parallel, the angles are equal. Note also that the angles marked a8 and b8 are vertically opposite angles so they are equal. Extra example 7 Find the size of /PAB. x° 126° Working Reasoning x 1 126 5 180 x 5 180 2 126 x 5 54 The angles marked 1268 and x8 are allied angles between parallel lines. Allied angles are supplementary, that is, they add to 1808. Teaching note: the converse is also true Students have seen that when parallel lines are crossed by a transversal, alternate angles and corresponding angles are equal and allied angles are supplementary. The converse of this is also true: if alternate angles and corresponding angles are equal and allied angles are supplementary, then the lines must be parallel. Extra example 8 relates to this property. 266 06 MW7NC_TRB Final.indd 266 19/12/11 5:12 PM 6 Extra example 8 chapter Geometry and space 6.4 Are line segments AB and CD parallel? A C 51° 50° B D Working Reasoning The angles marked 508 and 518 are alternate angles. The alternate angles are not equal so line segments AB and CD are not parallel. Alternate angles formed by a transversal cutting across two lines are equal if the lines are parallel. exercise 6.4 Ans wer s Below are the answers to the questions in exercise 6.4 in the student book. 1 ● 2 ● 3 ● a b c d e f g h a 47 d d = 61, k = 119 g x = 101, y = 101 b 94 e 48 h a = 36, b = 144 c m = 152, n= 152 f 88 i a = 31, b = 149, c = 149 a 56 d 115 g 134 b 124 e e = 59, f = 59 h a = 57, b = 123, c = 57 c 143 f m = 42, n = 42 i w = 112, x = 68, y = 68, z = 112 267 06 MW7NC_TRB Final.indd 267 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 4 ● 5 ● 6 ● a b c d AB AB AB AB a c e g /ACH /BCG /MBA /BCH is is is is parallel to CD because allied angles add to 1808. not parallel to CD because allied angles add to 1828. parallel to CD because allied angles add to 1848. parallel to CD because allied angles add to 1808. b d f h /BCG /BCH /CBM /ABN,/MBC,/BCH,/GCD 358. Harry’s line of sight and the sea are parallel, so d8 and 358 are alternate angles, which are equal. exercise 6.4 1 ● additional questions Copy each of the following figures. On your drawing, label the sizes of all angles that you had to find in order to determine the value of the pronumeral, then write the value of the pronumeral. a b 118° m° w° 36° c d 28° 53° k° y° 50° 37° Ans wer s 1 ● a b c d m = 36 w = 62 y = 90 k = 78 268 06 MW7NC_TRB Final.indd 268 19/12/11 5:12 PM 6.5 Triangles Teaching note: triangles as rigid shapes The student text at the beginning of this section shows a triangle made from plastic geostrips and photographs of triangles used in constructions because of their rigidity. It is worth getting students to experience the rigidity of a triangle compared with a quadrilateral (which can be pushed into different shapes). Plastic geostrips or rolls of newspaper taped together can be used successfully for this. Plastic geostrips are slightly ‘bendy’, so even though the triangles are rigid in the plane, the slightly flexible plastic strips can be distorted by heavy-handed students who may then miss the point of the activity! In this case, the rigidity of stiff rolls of newspaper may be better. Alternatively, if your school has a dome kit, components of the kit could be used. Construction of the entire dome would then make a worthwhile additional activity. The rigidity of the triangle may be compared with quadrilaterals, which can be made rigid by joining a diagonal (see beginning of section 6.6). Diagonal Teaching note: the angles in a triangle Class activity Sum of the angles of a triangle Tearing the corners from a triangle and arranging them to form a straight line is an excellent visual reinforcement for the concept that the three angles of a triangle add to 180°. It is advisable that students tear, rather than cut, the corners from their triangle. If they cut the corners, they may become confused about which is the cut edge and which are the arms of the angle of the triangle. If they make a second copy of their triangle, students can paste the triangle and the arranged pieces into their mathematics books. Students are sometimes not given the opportunity to progress from this teaching demonstration to a reasoned mathematical argument. It is important that students do not think the corner tearing demonstration actually proves the relationship. The following diagram (also included in the student book, shows that if we construct a line parallel to one side of the triangle, the same arrangement of the three angles to make a straight line can be used. This time, though, we are making use of alternate angles between parallel lines cut by a transversal to prove the relationship. Proof in this sense is an answer to the question ‘Why do the angles add to 180°?’ 269 06 MW7NC_TRB Final.indd 269 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book GeoGebra: angle sum of a triangle Angle sum of a triangle The interactive GeoGebra file, Angle sum of a triangle, is included in the student and teacher ebooks. The vertices of the triangle can be dragged to see that the three angles always add to 180°. Again, this is not a proof, but a very convincing demonstration. Clicking on the Proof checkbox displays the parallel line and gives students the opportunity to reason why the relationship is true. 270 06 MW7NC_TRB Final.indd 270 19/12/11 5:12 PM 6 GeoGebra: exterior angle of a triangle Exterior angle of a triangle chapter Geometry and space 6.5 The interactive GeoGebra file, Exterior angle of a triangle, is included in the student and teacher ebooks. The vertices of the triangle can be dragged to see that the three angles always add to 180°. After the students have had a chance to find the angle relationship for themselves, clicking on the check box displays the relationship and students are asked to explain it. In the triangle shown, we know that /ABC 1 /BCA 1 /CAB 5 180° (three angles of the triangle). But we also know that /ABC 1 /ABD 5 180° (straight line). So /BCA 1 /CAB must equal /ABD. Using pronumerals for the angles may make the explanation clearer for students. a° b° d° c° a 1 b 1 c = 180 (angles of a triangle add to 1808) d 1 c = 180 (adjacent angles that make a straight line add to 1808) So a 1 b must equal d. GeoGebra: types of triangles Triangles The interactive GeoGebra file, Triangles, is included in the student and teacher ebooks in both GeoGebra and HTML formats. Three sliders allow the side lengths of the triangle to be changed. Students can see the side and angle properties of the different types of triangles and can also see that there are some sets of lengths for which it is not possible to make a triangle. The check box Show type can be clicked off to give students the opportunity to name the triangle type themselves. 271 06 MW7NC_TRB Final.indd 271 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Teaching note: can any three side lengths make a triangle? Triangles The Triangles GeoGebra construction identifies side lengths that will not allow a triangle to be constructed. Making triangles with geostrips as shown on page 293 in the student book reinforces students understand why any three side lengths will not necessarily make a triangle. If pairs of students are given a random set of three geostrips (with three paper fasteners) some will find they can make triangles and others will not. This leads to a discussion of why some students were unable to make a triangle. Class activity: constructing special triangles The class activity Constructing special angles is available in the teacher ebook as a student worksheet. Class activity Constructing special triangles Required equipment: sharp pencil, ruler, pair of compasses Construction 1 This construction encourages students to think about triangle properties. Because they have used the same compass opening for two sides of their triangle, the triangle must be isosceles. Construction 2 In this construction, the same compass opening is used for all three sides of the triangle so the triangle must be equilateral. This is then linked with the angles which must, of course, be 608. These constructions could also be completed using interactive geometry software such as Geogebra. The circle tool then takes the place of compasses. 272 06 MW7NC_TRB Final.indd 272 19/12/11 5:12 PM Constructing special triangles 6 chapter Geometry and space 6.5 Construction 1 a In your mathematics workbook, draw a line segment 10 cm long. ■ Label the ends A and B. Using your ruler to measure, open your compass to 12 cm. ■ Place the compass point at A and draw an arc. ■ Place the compass point at B and draw another arc to intersect (cross) the first arc. 12 cm ■ A A B A B 10 cm B 10 cm 10 cm ■ Label the intersection point C. ■ Join each end of AB to C to form a triangle. Label the lengths of sides AC and BC. C A 10 cm B b What type of triangle have you constructed? c Measure the three angles of your triangle and label them on the triangle. Construction 2 Start with a 10 cm segment as in construction 1. ■ Label the ends of the segment D and E. ■ Place the compass point at the D and open the compass so that the pencil is exactly at E. Draw an arc. D 10 cm E D 10 cm E D 10 cm E F D E 273 06 MW7NC_TRB Final.indd 273 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book ■ Now do the opposite—place the point at E and place the pencil D. Draw another arc to intersect the first arc. Label the intersection point F. ■ Join each end of DE to F to make a triangle. d What sort of triangle is it? Can you explain why? e Carefully measure each of the angles of your triangle. If you have constructed your triangle accurately, each of the three angles should be 608. How close to 608 are your angles? Explain why the angles are 608. Answers a No answer required. b Isosceles triangle c Students should find their angles close to 658, 658 and 508. (Note: by calculation, the angles are approximately 65.388, 65.388 and 49.248.) d The triangle is an equilateral triangle because the length of two sides is determined by arcs that are equal in length to the 10 cm segment which forms the other side of the triangle. e The three angles are equal and must add to 1808. exercise 6.5 Ans wer s Below are the answers to the questions in exercise 6.5 in the student book. 1 ● 2 ● 3 ● 4 ● 5 ● 6 ● 7 ● a ^ABC (or ^CAB or ^BCA) c ^THA (or ^HAT or ^ATH) b ^BOX (or ^OXB or ^XBO) d ^ENP (or ^PEN or ^NPE) equilateral triangle: angles are 608, sides are equal a c e g i a a = 63 g g = 50 a d g j b d f h right-angled scalene triangle right-angled isosceles triangle obtuse-angled isosceles triangle obtuse-angled scalene triangle acute-angled isosceles triangle b b = 49 h h = 30 a = 32 x = 60, y = 60, z = 60 g = 60 d = 50, e = 80 a a = 65 g g = 70 b b = 77 h h = 110 a right-angled scalene triangle c c = 76 i i = 61 b e h k acute-angled equilateral triangle acute-angled isosceles triangle equilateral triangle (acute-angled) acute-angled scalene triangle d d = 36 j j = 90 b = 64 e = 74 h = 106 m = 45 c c = 90 i m = 44 e e = 86 k k = 30 c f i l d d = 108 j j = 39 f f = 53 l l = 108 c = 70, d = 40 y = 45, z = 75 t = 30, u = 120 x = 60, y = 60 e e = 125 k a = 75 f f = 52 l x = 70 b 274 06 MW7NC_TRB Final.indd 274 19/12/11 5:12 PM 8 ● a i b i C 8 cm A 5 cm d i 5 cm 6 cm A B ii acute-angled triangle iii equilateral triangle e i C 7 cm 5 cm 5 cm f i C 5 cm 5 cm A 3 cm B 7 cm ii acute-angled triangle iii isosceles triangle C 4 cm B 10 cm ii right-angled triangle iii scalene triangle C A 8 cm A ii acute-angled triangle iii scalene triangle c i 6.5 C 6 cm B 9 cm 6 chapter Geometry and space 5 cm A 8 cm B B ii right-angled triangle iii scalene triangle ii obtuse-angled triangle iii isosceles triangle g i h i C C 6 cm 12 cm 6 cm 5 cm A A B 8 cm B 13 cm ii right-angled triangle iii scalene triangle 9 ● 10 ● a c e g no; 5 cm 1 3 cm , 9 cm no; 2 cm 1 2 cm = 4 cm no; 5 cm 1 4 cm = 9 cm yes; 10 cm 1 3 cm . 11 cm a A ii acute-angled triangle iii isosceles triangle b d f h yes; 5 cm 1 6 cm . 7.5 cm yes; 3 cm 1 3 cm . 4 cm no; 6 cm 1 8 cm , 16 cm no; 7 cm 1 9 cm , 20 cm b A 7 cm 7 cm B 40° 6 cm 35° C i acute-angled triangle ii scalene triangle B 7 cm C i acute-angled triangle ii isosceles triangle 275 06 MW7NC_TRB Final.indd 275 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book c d A A 4 cm 8 cm B C 6 cm C 8 cm i acute-angled triangle ii isosceles triangle e 45° B 30° i obtuse-angled triangle ii scalene triangle f C C 4 cm A 8 cm 60° B 9 cm 55° A i obtuse-angled triangle ii scalene triangle g 30° 30° 8 cm M F 5 cm K i obtuse-angled triangle ii isosceles triangle 11 ● a 25° B 50° 8 cm i right-angled triangle ii isosceles triangle C 55° 40° 50° 6 cm 20° 6 cm A C 50° B i obtuse-angled triangle ii scalene triangle e A 5 cm B i obtuse-angled triangle ii isosceles triangle C 9 cm C 70° 40° 40° i right-angled triangle\ ii scalene triangle f C 40° C i right-angled triangle ii scalene triangle d A B A B i obtuse-angled triangle ii scalene triangle c L 5 cm b A B i acute-angled triangle ii scalene triangle h G E 10 cm A 20° 7 cm B i right-angled triangle ii scalene triangle 276 06 MW7NC_TRB Final.indd 276 19/12/11 5:12 PM g h R 35° P 55° 8 cm i right-angled triangle ii scalene triangle 12 ● 40° 40° Q 10 cm i obtuse-angled triangle ii isosceles triangle base 1.73 m, slope 2 m scale : 10 cm = 1 metre slope 1 metre slope measures 20 cm = 2 metres base measures 17.3 cm = 1.73 metres 30° base exercise 6.5 1 ● 6.5 R P Q 6 chapter Geometry and space additional questions Find values of the pronumerals in each of the following diagrams. a b c 8x° 64° 4x° 3x° 22° 34° y° w° d e z° 157° 2 ● 3 ● 62° 130° f 103° r° t° x° 126° This right-angled triangle has sides of length 3 cm and 4 cm as shown. a Measure the length of the third side of the triangle. 3 cm b For many centuries builders have known that a triangle with side lengths in these particular proportions is a right-angled triangle, and 4 cm they have used this fact to construct right angles when laying out the foundations for buildings. Suggest how you could use a ruler and a long piece of rope (for example, about 15 metres long) to mark out a right angle on the ground. If possible, try out your suggestion. Recent technology provides rugby players with instantaneous information about their chances of kicking a goal when they are at different distances from the goal. The player’s distance from the goal, the angle of vision to the goal posts (shaded) and the apparent width between the goals is displayed. Calculate the angle of vision, g8, for the situation shown in the diagram. 45° 19° g° 277 06 MW7NC_TRB Final.indd 277 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 4 ● 5 ● 6 ● Consider two triangles with the dimensions given. i Side lengths 7 cm, 5 cm, 4 cm ii Side lengths 14 cm, 10 cm, 8 cm a Use your ruler, pencil and compass to construct each of the triangles. b Notice that the triangles have the same shape but one is larger than the other. Measure the three angles in each triangle and write them on your constructions. c Suggest another set of three side lengths that would give a triangle with the same shape as these two triangles, but a different size. Consider two triangles with the dimensions given. i Side lengths 5 cm, 4 cm, 3 cm ii Side lengths 10 cm, 8 cm, 6 cm a Use your ruler, pencil and compass to construct each of the following triangles. b What sort of triangles are they? Harry and Kate are making a skateboard ramp. They want the ramp to make an angle of 30° with the ground and they want the top of the ramp to be exactly one metre above the ground. Harry and Kate have made a rough sketch of the side view of the ramp. They now need an accurate drawing of the side of the ramp so that slope they know how long to cut the wood for the base and the slope. 1 metre Choose a suitable scale (for example, 10 cm on your drawing 30° represents 1 m on the actual skateboard ramp) and make an base accurate diagram using your ruler and protractor. Measure the length of the base and the slope and work out the lengths Harry and Kate will need to cut for each. Ans wer s 1 ● 2 ● 3 ● 4 ● a w = 120 b x = 12 c y = 67 d z = 46 e r = 112 t = 68 f x = 23 a 5 cm b Knot or mark the rope into 3 m, 4 m and 5 m lengths and stretch into triangle shape. g 5 26 a and b i 102° ii 102° 5 cm 34° 4 cm 44° 10 cm 8 cm 34° 44° 14 cm 7 cm c Answers will vary. For example, 21 cm, 15 cm, 12 cm; 28 cm, 20 cm, 16 cm; 3.5 cm, 2.5 cm, 2 cm. 5 ● a i ii 3 cm 90° 90° 4 cm 6 cm 8 cm 5 cm 6 ● b right-angled triangles 10 cm Slope = 2 m Base = 1.73 m 278 06 MW7NC_TRB Final.indd 278 19/12/11 5:12 PM 6.6 Quadrilaterals Teaching note: angles in quadrilaterals Class activity Sum of the angles of a quadrilateral As for the angles of a triangle, the corner-tearing shown at the beginning of this section in the student book is a useful activity to reinforce the concept that the four angles of a quadrilateral add to 360°. Again, the visual demonstration can be extended to a consideration of why this is so. Dividing the quadrilateral into two triangles by drawing a diagonal readily shows that the 360° is made up of two lots of 180°. GeoGebra: angle sum of a quadrilateral Angle sum of a quadrilateral The interactive GeoGebra (and HTML) file Angle sum of a quadrilateral provides another, perhaps even more convincing, demonstration for the angle sum. Extra example 9 Find the values of the pronumerals in this rhombus. a° b° 56° c° Working Reasoning a 1 56 = 180 a = 124 A rhombus is a special parallelogram. Adjacent angles in all parallelograms are supplementary (add to 1808). b = 56 Opposite angles of parallelograms are equal. c = 124 a = 124 Opposite angles of parallelograms are equal. 279 06 MW7NC_TRB Final.indd 279 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Extra example 10 Identify each of these quadrilaterals. Draw a diagram for each to show the given information. a The quadrilateral has four right angles. One pair of opposite sides has length 12 cm and other pair of opposite sides has length 10 cm. b The quadrilateral has one pair of opposite sides parallel and no equal sides. Working Reasoning a The quadrilateral is a rectangle. The first sentence tells us that the quadrilateral is a rectangle (which includes squares). The second sentence tells us that it is a rectangle but not a square. 12 cm 10 cm 10 cm 12 cm b The quadrilateral is a trapezium. Trapeziums are the only quadrilaterals with just one pair of parallel sides. Teaching note: class inclusion Although we think of squares, rhombuses and rectangles as discrete shapes, they are all members of the family or class of parallelograms. They all satisfy the definition of a parallelogram as a plane shape with both pairs of opposite sides parallel. In the same way we can consider squares as special cases of both rhombuses and rectangles as ■ ■ a square is a special rectangle where all sides are equal. a square is a special rhombus where all angles are right-angles. GeoGebra: parallelogram family Parallelogram family The GeoGebra file Parallelogram family includes sliders for side lengths and angle. Students can make the special shapes of rhombus, rectangle and square simply changing the side lengths so that they are all equal, or by changing the angles to 908. Clicking on the Check box displays the special parallelogram name. 280 06 MW7NC_TRB Final.indd 280 19/12/11 5:12 PM 6 Class activity: how does a folding umbrella work? chapter Geometry and space 6.6 Required equipment: class set of umbrella ribs (three old umbrellas), GeoGebra file Umbrella Class activity Umbrella How does a folding umbrella work? If you have an old folding umbrella, it is worth pulling it apart to separate the eight ribs. Three old umbrellas gives a class set! The construction of each rib is based on a parallelogram and students can move the hinged parallelogram to see how the shape remains a parallelogram, even though the angles change. Umbrella a What special quadrilateral is formed by the hinged shape in the umbrella rib? b How is the hinged shape constructed so that this special quadrilateral is formed? c What important function do you think this special quadrilateral plays in the opening and closing of the umbrella? Ans wer s a parallelogram b The ribs have been constructed so that both pairs of opposite sides are equal. This ensures that the shape is a parallelogram and, therefore, that both pairs of opposite sides stay parallel. Note that this is the converse of the statement that a plane shape with both pairs of opposite sides parallel also has both pairs of sides equal. c This ensures that the umbrella folds neatly, with the folded parts moving parallel to each other as the umbrella is folded. Class activity: hinged quadrilaterals Required for this activity: GeoGebra or HTML files Toolbox and Car jack Class activity Hinged quadrilaterals Optional: several car jacks of the type shown below and an expanding box for tools, sewing or fishing. Although rectangles are probably the most common quadrilateral (for example in books, tables, floors and windows), the special properties of the rhombus make it extremely useful in the design of tools and other items. 281 06 MW7NC_TRB Final.indd 281 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Car jack Car jack Car jacks of this type are readily available from ‘op shops’ or car wreckers’ yards. Based on a rhombus, the jack hinges neatly and compactly. But the particular rhombus property that is important for the operation of the jack is that the diagonals of a rhombus are perpendicular. The screw thread represents one of the diagonals of the rhombus. The design of the base of the jack ensures that this diagonal remains horizontal. The interactive GeoGebra file Car jack demonstrates how the other (invisible) diagonal remains perpendicular to the base, ensuring that the car will be lifted vertically, perpendicular to the ground. Expanding toolbox Expanding toolbox The design of the expanding sewing box shown here is also found in tool boxes and fishing boxes. The interactive GeoGebra file Expanding box shows how the trays stay parallel to the base as they are pulled out. The brackets attached to the drawers form parallelograms that stay parallelograms because their opposite sides are of fixed equal lengths. There may be a student in the class who could bring a similar box to demonstrate how it works. 282 06 MW7NC_TRB Final.indd 282 19/12/11 5:12 PM 6 Hinged quadrilaterals chapter Geometry and space 6.6 1 Car jack a What is a car jack designed to do? Open the GeoGebra file Car jack. Drag point P to simulate operating the car jack. b What special quadrilateral is formed by the four equal sides of the quadrilateral? c What property of this special quadrilateral is useful for storing the jack? d The horizontal screw thread of the jack represents one of the diagonals of the quadrilateral. What do you know about the two diagonals of this special quadrilateral? e Why is this property useful for the way the jack is designed to work? 2 Toolbox a What special quadrilateral shape is formed by the hinged brackets connecting the trays? b How is the box constructed so that this special quadrilateral shape is formed? c Can you explain why the trays stay parallel to each other (and to the base of the toolbox) as they lift out? Ans wer s 1 Car jack a A car jack is designed to lift a car to change a wheel. b rhombus c The sides are equal, so it can be folded flat. d The diagonals are at right angles (perpendicular). e The care moves at right angles to the ground, that is, vertically upwards. 2 Toolbox a A parallelogram (or in this case the parallelogram may be a rhombus) b The important aspect of the design is that the bracket strips are connected to the trays so that a parallelogram is formed. c The design is using the property that a shape with both pairs of opposite sides equal will also have both pairs of opposite sides parallel. Teaching note: exercise 6.6 question 9 Class activity Hinged quadrilaterals Scissor lift Question 9 relates to the scissor lift which incorporates the rhombus properties of parallel and equal sides for compactness of closing. Again, as in the case of the car jack, the perpendicular diagonals of the rhombus ensure that the hinged vertices of the parallel bars move vertically, perpendicular to the ground. This ensures that the work platform remains horizontal. Students could consider whether any parallelogram would work, or whether it is important that the special case rhombus is used. If possible, it is worth getting students to construct the scissor lift model from geostrips and paper fasteners. They enjoy the tactile feel of opening and closing the model while observing how the sides remain parallel and the pivot points move up and down perpendicular to the horizontal. One previously disengaged student became totally involved with this activity. He had been sitting operating the rhombus geostrip linkage for a short time then suddenly announced: ‘My dad drives one of those scissor lifts’. This one experience seemed to generate an interest in mathematics for this student. Students also enjoy operating the virtual scissor lift in the interactive GeoGebra file Scissor lift. 283 06 MW7NC_TRB Final.indd 283 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Scissor lift What property of the rhombus makes it useful in the scissor lift? Move point P horizontally. exercise 6.6 Ans wer s Below are the answers to the questions in exercise 6.6 in the student book. 1 ● 2 ● 3 ● 4 ● 5 ● a a = 55 g g = 108 b b = 49 h h = 114 a square d rectangle g parallelogram c c = 124 i i = 110 d d = 113 j j = 132 b rhombus e parallelogram h square e e = 103 k k = 58 f f = 56 l l = 82 c trapezium f kite i rhombus rectangle, trapezium, square a d g j m k = 93 d = 69 m = 110 x = 118, y = 62, z = 118 a = 121, b = 121 a b e h k n p = 90, q = 90, r = 90 e = 101, f = 50 u = 40, v = 140 p = 40, q = 140, r = 40 d = 70 c f i l o t = 45 x = 62, y = 118 m = 72, n = 108 x = 95, y = 97 a = 125, b = 60 5.4 cm 114° 66° 4 cm 5.4 cm 66° 114° 5.4 cm b Both pairs of opposite sides of a parallelogram are parallel and equal. c The opposite angles of a parallelogram are equal. The adjacent angles of a parallelogram add to 1808. 284 06 MW7NC_TRB Final.indd 284 19/12/11 5:12 PM 6 ● 7 ● 6 a Yes, they all have both pairs of opposite sides parallel and equal. b A rhombus is a parallelogram with four equal sides. A rectangle is a parallelogram with four right angles. A square is a parallelogram, with four equal sides and four right angles. chapter Geometry and space 6.6 a, c 90° 3.3 cm 3.3 cm 102° 102° 4.2 cm 4.2 cm 66° axis of symmetry b The kite has two pairs of equal sides and one pair of equal angles. 8 ● a b two 74° 114° 66° 9 ● 10 ● 11 ● 12 ● 13 ● 14 ● 105° 74° 75° 106° 106° 78° 102° 90° 90° a rhombus b The bars stay parallel to each other. Both pairs of opposite sides of the rhombus are parallel and equal, so the bars close up neatly and compactly. The diagonals of each rhombus are at right angles to each other so the rhombuses move up and down (or in and out) at right angles to the base. This is particularly important in the case of the scissor lift to ensure that the work platform remains parallel to the ground. b /DCB 5 90° /CBA 5 110° /BAD 5 50° a kite a trapezium b It is the best shape to fit a bicycle with the least amount of wasted space. The two trapeziums fit beside each other to make a rectangle. 608 and 1208 h = 72, s = 74 a Examples are shown below. i ii iii 285 06 MW7NC_TRB Final.indd 285 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book b No. The angles in a quadrilateral always add up to 3608, so if you had three right angles then the fourth angle would also have to be a right angle. 15 ● 16 ● a a = 29 b b = 120 c c = 123, d = 57 James is correct. Both a square and a rhombus have both pairs of opposite sides parallel and all sides equal. They also both have opposite angles equal. The difference is that for a square all the angles must be 908, but for the rhombus there is no such limitation. exercise 6.6 1 ● 2 ● additional questions a Using your ruler and protractor, carefully draw a quadrilateral which has exactly i four right angles. ii two right angles. iii one right angle. b Is it possible to draw a quadrilateral that has exactly three right angles? Explain. Find the value of the pronumerals in each of the following figures. a b c 72° a° 42° 23° d 57° 67° 94° 3 ● d x = 106 92° d° 65° b° c° 59° 77° x° When a square is folded in half from corner to corner as shown, two triangles are formed. What are the sizes of the three angles in each triangle? Ans wer s 1 ● a i ii iii b No, because three right angles add to 2708, leaving 908 for the fourth angle, so there would be four right angles. 2 ● 3 ● a a = 29 b b = 120 c c = 123, d = 57 d x = 106 458, 458, 908 286 06 MW7NC_TRB Final.indd 286 19/12/11 5:12 PM 6.7 Representing three-dimensional objects in two dimensions Teaching note: oblique and isometric drawings The advantages and disadvantages of oblique and isometric representations can be discussed with students. The oblique drawing has the advantage that front and back faces are accurately depicted, but the disadvantage is that the depth cannot be accurately shown. The isometric drawing has the advantage that all three dimensions—length, width and depth—can be shown accurately, but the drawing appears distorted to our eye because we are accustomed to a perspective view. Extra example 11 Use 1 cm graph paper to construct an oblique drawing of a box with a front face of 9 cm by 7 cm. Working Reasoning Draw a rectangle 9 cm by 7 cm, then draw line segments at 458 to the horizontal from three vertices of the rectangle. The length of these segments depends on how deep you wish the box to appear. Draw the remaining horizontal and vertical line segments to complete the box. Extra example 12 Use isometric grid paper to construct an isometric drawing of a box 20 mm high with a base 50 mm by 20 mm. continued 287 06 MW7NC_TRB Final.indd 287 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Extra example 12 continued Working Reasoning Draw a vertical line segment 2 units long to represent the height of the box. Draw segments 5 and 2 units long to represent two edges of the base of the box. Draw the remaining line segments to complete the box, making sure the vertical edges are all 2 units long. The box could have been drawn the other way around, with the 5 unit segment to the left and the 2 unit segment to the right. Teaching note: plans and elevations There is an excellent isometric drawing tool provided at the National Council of Teachers of Mathematics website at www.nctm.org. Extra example 13 Use this isometric drawing to draw the a plan. b front elevation. c side elevation. Front continued 288 06 MW7NC_TRB Final.indd 288 19/12/11 5:12 PM 6 Extra example 13 continued chapter Geometry and space 6.7 Working Reasoning a The plan shows the ‘floor’ area that the shape covers. b The front elevation shows what we would see if we looked directly at the front of the shape without seeing the depth. It appears as if the blocks have been moved forward so that they are all in the same plane. c The side elevation shows what we would see if we looked directly at the side of the shape without seeing the depth. exercise 6.7 Ans wer s Below are the answers to the questions in exercise 6.7 in the student book. 1 ● 2 ● 3 ● 4 ● 289 06 MW7NC_TRB Final.indd 289 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 5 ● a i ii iii 4 cm 5 cm 4 cm 4 cm 5 cm 4 cm b i ii iii 2 cm 3 cm 2 cm 4 cm 3 cm 4 cm c i ii iii 2 cm 2 cm 4 cm 5 cm 4 cm 5 cm d i ii 4 cm 3.5 cm 1.5 cm 6 ● iii 3.5 cm 1.5 cm 4 cm a i ii iii b i ii iii c i ii iii d i ii iii 290 06 MW7NC_TRB Final.indd 290 19/12/11 5:12 PM 7 ● e i ii iii f i ii iii a i ii iii 2.5 cm 6.7 3 cm 3 cm 6 cm 6 chapter Geometry and space 4 cm 6 cm 4 cm b i ii iii 25 cm 25 cm 40 cm 50 cm 40 cm 50 cm 8 ● 4m 5m 3.5 m 9 ● 10 ● 11 ● a b c d Answers will vary according to prisms used. Check with your teacher. a plan b side elevation c isometric drawing d oblique drawing 291 06 MW7NC_TRB Final.indd 291 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book exercise 6.7 additional questions 1 ● Make an isometric drawing of a box 2 units high, 4 units wide and 6 units long. 2 ● Make an oblique drawing of a box 2 units high, 4 units wide and 6 units long. 3 ● Use this isometric drawing to draw the a plan. b front elevation. c side elevation. Front 292 06 MW7NC_TRB Final.indd 292 19/12/11 5:12 PM Ans wer s 6 chapter Geometry and space 6.7 1 ● 2 ● 3 ● a b plan c front elevation side elevation 293 06 MW7NC_TRB Final.indd 293 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Analysis tasks Answers to the student book task, A parking problem, are included in this section together with two additional analysis tasks: Polyominoes and Boom angles. Student book: a parking problem A parking problem links to the chapter warm-up worksheet Parallel and angle parking and looks at how angles are important in safe street (kerbside) parking. Students will discover that the greater the angle the cars make with the kerbside, the greater the number of cars that can fit. However, there are many other issues to take into account, such as the safety of cars backing out in front of traffic. Discussion of these issues could be at whole class level or students could work in groups. BLM A parking problem The teacher ebook includes a blackline master template (A parking problem) of 1:100 scale parking space models of the 2.7 3 5.5 m spaces for angle parking and the slightly longer 2.7 3 6 m spaces for parallel parking. It is easier if the two different sizes of parking space are photocopied onto different coloured paper. These pieces could be saved and used again by other classes to save class time—in this case, thin card may be preferable. The pieces could also be laminated for future use. For the slightly longer parking spaces for parallel parking, each group should be given at least 10 pieces. (Actually, only six are needed, but we don’t want to give them the answer before they start!). For the angle parking spaces, each group should have at least 30 pieces so that they can set out the 908 and 608 parking at the same time. A nss w e rs An a Parallel parking: 6 cars b Spaces can be shorter for 608 or 908 angle parking because there is space in the road behind the parked cars, so cars do not need extra space for getting in or out of the parking space. c 908 angle parking: 14 cars d 608 angle parking: 12 cars Note: Students may also like to see how many cars would fit for 458 angle parking. 458 angle parking: 9 cars 294 06 MW7NC_TRB Final.indd 294 19/12/11 5:12 PM e Advantage Disadvantage Parallel parking Parallel parking is suited to narrower roads and is safer for cars leaving the parking space. Fewer cars will fit. Angle parking More cars can fit in a certain length of roadway than for parallel parking. 458 and 608 parking fit more cars than parallel parking, but require less road width than 908 parking. Angle parking spaces extend out further into the roadway, and it is harder for drivers to see traffic coming when they are backing out of an angle parking space. 6 chapter Geometry and space f Factors affecting choice of parking type include ■ width of the road ■ number of parking spaces needed ■ length of the available parking strip ■ amount of traffic on the road, which could make angle parking more dangerous ■ depth of gutters in many towns, where parallel parking would not allow car doors to be opened on the passenger side ■ how busy the area is in terms of parking requirements. Additional task: polyominoes This analysis task develops spatial visualisation skills. The task starts with triominoes, shapes made from three squares joined by sides, then leads on to tetrominoes and pentominoes. Students are asked to work out how many different tetrominoes and pentominoes are possible, drawing each of the possible arrangements of squares. They are then asked to use the shapes to fill certain regions. The game of dominoes has rectangle shaped pieces made up of two squares joined together. We could join together different numbers of squares to make triominoes (three squares), tetrominoes (four squares) and pentominoes (five squares). If we have two squares there is only one way of putting them together, as shown above. If we imagine that we can pick up the triominoes and turn them over (reflect them) and turn them around (rotate them), there are two different ways of putting three squares together. These two triominoes are the same The two different triominoes 295 06 MW7NC_TRB Final.indd 295 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book a Fill this 6 3 3 rectangle using a combination of the two different triominoes. b There are five different tetrominoes. Draw the five tetrominoes, being careful to check whether some of your tetrominoes are actually the same. c Fill this 8 3 3 rectangle using i two different tetrominoes. ii four different tetrominoes. d There are 12 different pentominoes. Draw the 12 tetrominoes, again being careful to check whether some are actually the same. e Fill this 10 3 3 rectangle using i two different pentominoes. ii four different pentominoes. f Fill this square with five different pentominoes. An s w e rs a b 296 06 MW7NC_TRB Final.indd 296 19/12/11 5:12 PM c i 6 chapter Geometry and space ii d e i ii f 297 06 MW7NC_TRB Final.indd 297 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Additional task: boom angles This task uses the angles displayed on the boom of a crane together with the safe loads and the horizontal reach of the crane. The task gives practice in drawing angles and in interpreting information in a table. A Boom angles The part of a crane that can be tilted at different angles is called the boom. The angle, /ABC, that the boom makes with the horizontal is called the boom angle. a Use your protractor to find the boom angle for this crane. Boom B C ° 90 0° Cranes have an angle measurer resembling a 908 protractor on the boom which tells the crane driver the angle the boom makes with the horizontal. 80 ° 70° 0° ° 2 60° 50 ° 40° 30 10 ° The crane shown in the photographs below was working on a building construction site and was lifting concrete walls into position. The boom is shown in two different positions. Position 1 Position 2 b Measure the boom angle (marked) for the crane in the two different positions, and compare your answers with the crane’s angle measurer in each position. 298 06 MW7NC_TRB Final.indd 298 19/12/11 5:12 PM c The boom is telescopic and its length can be changed, depending on how far it must reach to set down its load. If the boom is fully extended and the angle is too small, the crane may tip over with very heavy loads. The capacity chart in the cabin tells the driver the boom angle that is safe for a particular load, and how far the crane can reach horizontally for that angle and that boom length. The capacity chart on the right shows the loads that can be safely lifted for different boom angles when the boom is fully extended to 31.5 m. It also shows how far the crane can reach for each angle. For each of the two positions of the crane in part b, use the capacity chart to find the maximum load the crane could safely lift. d What load could the crane safely lift if the boom angle is 408? e For each of the two positions of the crane in part b, use the capacity chart to find how far the boom could reach horizontally. f Using your protractor and ruler, make a careful drawing of the boom to show the angle it would need to make with the horizontal for a load of 8100 kg. 6 chapter Geometry and space Horizontal reach Boom length 31.5 m Boom angle (8) Load (kg) Horizontal reach (m) 79.0 10 000 6 77.0 10 000 7 75.0 10 000 8 73.5 9000 9 71.5 8100 10 67.5 6550 12 63.5 5000 14 59.0 4000 16 55.0 3200 18 50.0 2500 20 45.5 1800 22 40.0 1300 24 33.5 900 26 25.5 650 28 An A n s w e rs a 44° b position 1 = 48°; position 2 = 80° (so the crane’s measurer is wrong in both positions: it says 53° instead of 48° in position 1 and 51° instead of 80° in position 2) c position 1 = 2200 kg; position 2 = 10 000kg d 1300 kg e position 1 = 21 m; position 2 = 6 m f 71.5° Further investigations in chapter 15 The following investigations in chapter 15 are suitable as additional or alternative tasks for chapter 6. ■ ■ Catching the sun’s heat Star polygons The investigation Catching the sun’s heat involves drawing angles accurately with a protractor and interpreting given data to show the ideal tilt angles for solar panels in various places in Australia. The investigation Star polygons builds on students’ familiarity with a 5-pointed star to look at other star polygons. The task incorporates fractions as well as interesting geometry. A blackline master file in the student and teacher ebooks provides the templates for the vertices. 299 06 MW7NC_TRB Final.indd 299 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Review Geometry and space Visual map suggestion Lines, segments and rays Perpendicular At right angles Parallel Same distance apart; in the same plane and never meet Acute < 90˚ Right 90˚ Obtuse ⬎ 90˚ ⬍ 180˚ Parallel lines Alternate angles are equal Corresponding angles are equal Allied angles are supplementary Straight 180˚ Revolution 360˚ Vertically opposite angles are equal Angles Complementary angles add to 90˚ Reflex ⬎ 180˚ ⬍ 360˚ Supplementary angles add to 180˚ Triangles 3 sides Sum of angles = 180° Sides Scalene Isosceles Angles Equilateral Acuteangled Obtuseangled Rightangled 300 06 MW7NC_TRB Final.indd 300 19/12/11 5:12 PM 6 chapter Geometry and space Quadrilaterals 4 sides Sum of angles = 360° Special Quadrilaterals Parallelograms Both pairs of opposite sides parallel and equal. Adjacent angles supplementary. Diagonals bisect each other. Trapezia 1 pair of opposite sides parallel. 2 pairs of supplementary allied angles. Rectangles All angles 90°. Diagonals equal in length. Squares All angles 90° and all sides equal. Diagonals equal and intersect at right angles. Kites Two pairs of adjacent sides equal. One pair of opposite angles equal. Diagonals intersect at right angles. Rhombuses All sides equal. Diagonals intersect at right angles. Note that the boxes for rectangles, squares and rhombuses have been included in the box for parallelograms to indicate that they all belong to the family of parallelograms. Similarly squares have been shown overlapping the boxes for rectangles and rhombuses. Revision answers 1 ● 6 ● 7 ● 8 ● 2 ● C E 3 ● C 4 ● B 5 ● E a The line PQ is parallel to the line RS. b i AB is perpendicular to CD. ii EF is parallel to GH. a 144° a i b i b 215° /MLG or /GLM /PHC or /CHP ii acute angle ii reflex angle iii 65° iii 280° 9 ● D A C B 301 06 MW7NC_TRB Final.indd 301 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 10 ● E C D A 11 ● 12 ● B a complementary: 53° and 37° b supplementary: 139° and 41° a b 45° E 135° C 13 ● 14 ● 15 ● 16 ● 17 ● 18 ● 19 ● D a k 5 28 b /EXY 5 73° a a 5 50 (cointerior angles in parallel lines add to 1808) b b 5 146 (corresponding angles in parallel lines are equal) a a 5 124 b b 5 22 a rectangle; a 5 30 d square; d 5 90 b parallelogram; b 5 98 e rhombus; e 5 105 c kite; c 5 115 f trapezium; f 5 90, g 5 130 a Yes, the sum of the two shortest sides is more than the third side, so the sides will meet. b Yes, the sum of the two shortest sides is more than the third side, so the sides will meet. c No, the sum of the two shortest sides is equal to the third side, so the sides will not meet. a p 5 30 a b 1508 A 5.5 cm B 20 ● c a 5 80; b 5 100; c 5 43 c 4 pieces d 908 b isosceles c acute-angled b isosceles c right-angled 6 cm C 6 cm a F 5 cm D 5 cm E 302 06 MW7NC_TRB Final.indd 302 19/12/11 5:12 PM 21 ● a b scalene c obtuse-angled L 30° J 22 ● i 23 ● i 6 chapter Geometry and space 40° K 8 cm ii 24 ● 303 06 MW7NC_TRB Final.indd 303 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book Practice quiz answers Practice quiz Chapter 6 1 ● 6 ● 11 ● C A D 2 ● 7 ● 12 ● 3 ● 8 ● 13 ● E A A 4 ● 9 ● 14 ● E D C 5 ● 10 ● 15 ● C A D D B B Chapter tests Test A Chapter test A Chapter 6 Multiple-choice questions 1 ● 2 ● 3 ● 4 ● 5 ● Two lines which are at right angles to each other are A adjacent. B perpendicular. C complementary. D parallel. E E vertically opposite. The size of /AFD is A 378 B 538 C 1278 D 1438 E 2338 A D C F 37° B Which one of the following is a pair of complementary angles? A 818 and 98 B 908 and 1808 C 648 and 1168 D 778 and 238 E 608 and 208 Which one of the following statements is not correct? A Two line segments that are parallel could not be perpendicular. B Two lines that are in the same plane and do not meet must be parallel. C Complementary angles add to 908. D Vertically opposite angles are always equal. E Adjacent angles are always supplementary. Which of the following sets of three lengths could be the sides of a triangle? A 5 cm, 8 cm, 15 cm B 6.5 cm, 6.5 cm, 13 cm C 8.1 cm, 9.8 cm, 17.9 cm D 6.2 cm, 12.4 cm, 18.6 cm E 11.5 cm, 8.1 cm, 16.7 cm [5 3 2 = 10 marks] 304 06 MW7NC_TRB Final.indd 304 19/12/11 5:12 PM chapter 6 Geometry and space Short-answer questions 6 ● 7 ● Carefully draw diagrams using a ruler and a pencil, then add the correct symbols to show a two intersecting line segments that are perpendicular to each other. b two lines that are parallel to each other. [1 1 1 = 2 marks] Find the size of the following angle and state its type. 50 280 270 260 250 12 01 30 24 02 30 3 50 350 3 40 3 30 3 40 30 01 20 40 110 250 100 90 80 70 60 260 270 280 290 50 340 10 20 13 02 3 0 22 23 350 0 10 40 30 190 20 02 15 01 3 20 170 180 170 1 60 10 03 30 8 ● 160 190 200 10 02 10 2 0 01 40 290 22 20 0 30 80 90 100 110 14 10 30 70 60 S P W A Y Road Philip Street, Hamilton Road, Bay Road and Market Street meet at a junction as shown. a What is the size of the acute angle that Philip Street makes with Hamilton Road? b Which two roads or streets form a pair of vertically opposite angles? c What is the size of the acute angle that Market Street makes with Hamilton Road? reet 76° Ba 24° t ee tr tS e ark M 43° St Philip yR oad 9 ● [1 1 1 1 1 1 1 = 4 marks] Hamilton a Measure the size of /WAY. b Name the angle that is the supplement of /WAY. [1 1 1 1 1 = 3 marks] 305 06 MW7NC_TRB Final.indd 305 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 10 ● A a b c d pair of parallel lines is cut by a transversal as shown. Name a pair of alternate angles. Name a pair of corresponding angles. A Name a pair of cointerior angles. What is the size of /DGH? C H 143° G D F E B [1 1 1 1 1 1 1 = 4 marks] 11 ● 12 ● A quadrilateral has two sides of length 7 cm and two sides of length 10 cm. Jason says that the quadrilateral must be a rectangle. Draw diagrams to show that there are two other possibilities. In each case name the special quadrilateral you have drawn. [2 1 2 = 4 marks] Find the value of the pronumerals in each of the following. a b 43° 68° b° 3b° a° 55° c d b° 82° 48° 125° d° e f a° 65° x° 145° 112° 40° g h 104° 80° a° 98° a° [8 3 2 = 16 marks] 306 06 MW7NC_TRB Final.indd 306 19/12/11 5:12 PM 6 chapter Geometry and space Extended-response questions 13 ● 14 ● Follow these steps. a Use your compass, ruler and pencil to construct ^ABC with the following sides: BC = 7 cm, AB = 3.5 cm and CB = 5 cm. b Classify ^ABC according to its side lengths. c Classify ^ABC according to its angle sizes. [2 1 1 1 1 = 4 marks] Follow these steps. a Starting with the edge shown, construct an isometric drawing of 5 cm cube. (Assume the isometric grid is a 1 cm grid.) b Draw a front elevation of this three-dimensional shape. [2 1 1 = 3 marks] [Total = 50 marks] Te s t A ans wer s 1 ● 6 ● 7 ● 8 ● 9 ● 10 ● 2 ● B a 3 ● D A 4 ● D 5 ● E b 2578, reflex a 298 b /PAW a 808 b Hamilton Road and Philip Street a b c d c 578 /AFG and /DGF or /BFG and /CGF /AFG and /CGH or /BFG and /DGH or /EFB and /FGD or /EFA and /FGC /AFG and /CGF or /BFG and /DGF 378 307 06 MW7NC_TRB Final.indd 307 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book 11 ● b parallelogram kite 7 7 10 10 10 7 7 10 12 ● 13 ● a b 5 28 e a 5 105 a b a 5 82 f x 5 103 A 3.5 cm B 14 ● c b 5 42 g a 5 76 d d 5 27 h a 5 82 b Scalene c Obtuse-angled 5 cm 7 cm a C b Test B Chapter test B Chapter 6 Multiple-choice questions 1 ● Two angles that add to 908 are A complementary. B supplementary. C acute. D obtuse. E reflex. 308 06 MW7NC_TRB Final.indd 308 19/12/11 5:12 PM 2 ● 3 ● 4 ● 5 ● The value of x is A 108 B 508 C 1008 D 1408 E 2808 x° 6 chapter Geometry and space 37° 43° x° Which of the following is a pair of supplementary angles? A 1638 and 278 B 598 and 1218 C 418 and 498 D 2408 and 608 E 908 and 1808 Which one of the following statements is correct? A Adjacent angles are always supplementary. B Two line segments that intersect at 908 are parallel. C Vertically opposite angles are always complementary. D Two lines are parallel if they are in the same plane and they never meet. E Two line segments are parallel if they are in the same plane and they never meet. Which of the following could not be the sides of a triangle? A 3 cm, 4 cm, 5 cm B 8.2 cm, 9.8 cm, 17.9 cm C 8.1 cm, 9.9 cm, 17.9 cm D 5 m, 12 m, 16 m E 5 m, 8 m, 15 m [5 3 2 = 10 marks] Short-answer questions 6 ● Name a pair of a supplementary angles. b complementary angles. c vertically opposite angles. F E A O B C D [1 1 1 1 1 = 3 marks] 309 06 MW7NC_TRB Final.indd 309 19/12/11 5:12 PM MathsWorld 7 Australian Curriculum edition Teacher book a Find the size of the following angle. 0 270 280 29 50 26 02 40 21 170 1 60 350 190 200 340 15 01 30 350 3 40 3 30 3 22 00 60 30 290 70 280 270 260 250 80 90 02 24 20 100 11 01 1 40 03 50 30 1 40 10 2 0 10 02 0 20 31 190 20 0 0 1 10 20 180 31 50 30 170 1 30 00 60 40 160 03 0 90 80 70 10 10 1 20 03 02 23 32 20 4 02 50 7 ● 0 13 b State its type. 8 ● [1 1 1 = 2 marks] Measure the size of /POM. P Q O R M [1 mark] 9 ● Find the value of the pronumerals. a y° x° b 35° a° 2a° 3a° 310 06 MW7NC_TRB Final.indd 310 19/12/11 5:12 PM c 6 chapter Geometry and space d 130° x° c° 115° e 30° f 100° 40° y° 145° d° g P h Q 100° d° x° 95° 48° 79° S 60° R [8 3 2 = 16 marks] Extended-response questions 10 ● 11 ● 12 ● Follow these steps. a Use a compass and a ruler to construct a triangle with sides of length 5 cm, 5 cm and 7 cm. b Classify the triangle according to its side lengths. [2 1 1 = 3 marks] Follow these steps. a Use a pencil and a ruler to draw a diagram showing a line AB that is parallel to a line CD. Use the correct symbols to show that the lines are parallel. b On the same diagram, draw a line segment EF that cuts both AB and CD. c Label the point M where EF cuts AB and the point N where EF cuts CD. d Name a pair of alternate angles. e Name a pair of corresponding angles. f Name a pair of cointerior angles. [2 1 1 1 1 1 1 1 1 1 1 = 7 marks] Complete the following. a State two properties that rectangles, rhombuses, parallelograms and kites have in common. b Draw and name three different types of quadrilaterals with two sides of length 3 cm and two sides of length 5 cm. [2 1 3 = 5 marks] 311 06 MW7NC_TRB Final.indd 311 19/12/11 5:13 PM MathsWorld 7 Australian Curriculum edition Teacher book 13 ● Follow the steps. a Use the isometric grid paper below to make an isometric drawing of a box 3 units high, 4 units wide and 6 units long. Start with the edge shown. b Draw a front elevation of this 3-dimensional shape. [2 1 1 = 3 marks] [Total = 50 marks] Te s t B ans wer s 1 ● 6 ● 7 ● 8 ● 9 ● A 2 ● 3 ● C B 4 ● D 5 ● E a /FOA and /AOD or /FOB and /BOD or /FOC and /COD or /EOF and /FOC or /EOA and /AOC or /EOB and /BOC or /FOE and /EOD or /EOD and /DOC b /AOB and /BOC or /EOF and /FOA or /COD and /FOA c /EOF and /COD a 75° b acute 1358 a x 5 145,y 5 35 e y 5 45 b a 5 60 f d 5 70 c c 5 130 g x 5 110 d x 5 35 h d 5 42 312 06 MW7NC_TRB Final.indd 312 19/12/11 5:13 PM 10 ● a 6 chapter Geometry and space b isosceles 5 cm 5 cm 7 cm 11 ● a, b, c E M A B d /AMN and /DNM or /BMN and /CNM e /AME and /CNM or /BME and /DNM or /AMN and /CNF or /BMN and /DNF f /AMN and /CNM or /BMN and /DNM N C D F 12 ● a four sides; angles add to 3608 3 cm 3 cm b 3 cm 3 cm 5 cm 5 cm rectangle 13 ● a 5 cm 5 cm parallelogram kite b 313 06 MW7NC_TRB Final.indd 313 19/12/11 5:13 PM