MODULE 3: Spces in the Environment Unit 1 Angles The ojectives of this section re to recognise the reltionship etween turns nd ngles understnd how to mesure ngles nd how to work out their size in prticulr prolems. Angles re n importnt uilding lock in geometry nd trigonometry s you will see lter. In this unit you will see how turns re relted to ngles, how to mesure them nd how to work out their size in prticulr prolems. 1.1 Angles nd Turns You will need to understnd clerly wht the terms such s turn, hlf turn, etc. men in terms of ngles. There re 360 in one complete turn, so the following re true. 3 1 turn 4 turn 1 2 turn 1 4 turn is 360 is 270 is 180 is 90 You lso need to refer to compss points: (north (), south (S), est (E), west (W), northest (E), southest (SE), southwest (SW) nd northwest (W). W E Turning from to S is 180 clockwise or nticlockwise W E Turning from E to SE is 90 clockwise (or 270 nticlockwise) SW S SE Turning clockwise from E to E is 45 (or 315 nticlockwise) 1
1.1 Exmple Wht ngle do you turn through if you turn () from E to W nticlockwise, from E to clockwise Solution W E () You cn see tht this is 90 (or 1 4 turn). W E This is 3 4 turn, i.e. 270. SW S SE Exercises 1. John is stnding on hill. The church is north of the point where he stnds. Church TV Mst Chimney John Frm Ok Tree Bridge () In wht direction is he fcing if he looks t: (i) the chimney, (ii) the ridge, (iii) the TV mst, (iv) the frm, (v) the ok tree 2
1.1 Wht ngle does John turn through if he turns clockwise from looking t: (i) the church to the frm, (ii) (iii) (iv) (v) the ok tree to the ridge, the TV mst to the ok tree, the ridge to the TV mst, the TV mst to the church Wht would the ngles e for question if John turned nticlockwise insted of clockwise 2. In gme, you spin pointer nd let it stop. Wht ngle does the pointer turn through if it completes: () 1 turn, 2 turns, 3 4 turn, 1 1 4 turns, (e) 1 3 4 turns, (f) 2 1 4 turns 3. Wht ngle do you turn through if you turn clockwise from fcing: () to E, W to W, SE to W, E to, (e) W to E, (f) S to SW, (g) S to SE, (h) SE to SW, (i) E to SW 4. Wht ngle do you turn through if you turn nticlockwise from fcing: () to SW, S to SW, W to W, E to S 5. In wht direction will you e fcing if you turn: () 180 clockwise from E, 180 nticlockwise from SE, 90 clockwise from SW, 45 clockwise from, (e) 225 clockwise from SW, (f) 135 nticlockwise from, (g) 315 clockwise from SW 3
1.1 6. orvo stnds on low hill. The digrm elow shows some of the things he cn see. Using informtion from the digrm, nswer the following questions. () Wht is E of orvo Wht is SE of orvo (e) orvo turns from looking t the Old Fort to look t the ship. Wht ngle does he turn through Explin why there is more thn one nswer to this question. Wht ngle does orvo turn through if: (i) he turns clockwise from looking t the ship to the crne, (ii) he turns nticlockwise from looking t the rdio mst to the fctory, (iii) he turns nticlockwise from looking t the fctory to the ship orvo strts looking t the fctory. Wht does he end up looking t if he turns: (i) 135 clockwise, (ii) 270 nticlockwise, (iii) 225 clockwise, (iv) 405 clockwise Fctory Rdio Mst Lighthouse orvo igel W E Church Tower S Ship Crne Old Fort 7. Use the digrm on the next pge to nswer these questions. () Wht is of the shop Wht is W of the church 4
1.1 (e) (f) (g) (h) Wht is E of the church Wht is E of Big Rock nd E of the shop Wht is SW of Big Rock In wht direction should you wlk from the ech to get to the tower In wht direction should you wlk from the ech to get to the church If you wlk SE from the windmill, will you get to the tower Explin your nswer. Big Rock Church Windmill Lighthouse Shop Tower Bech 8. The sketched mp elow shows towns nd villges in west Jmic. Crien Se Dvis Cove Luce Green Islnd Silver Spring Grnge Hill egril Bnury Sheffield Little London Brighton Svnnh l Mr 5
1.1 () (e) (f) Wht is est of Silver Spring Wht is north of Grnge Hill Wht is SE of Grnge Hill Wht is W of Little London Wht is north est of Green Islnd nd north west of Bnury Wht is north west of Little London nd north of Sheffield 9. The sils of windmill complete one full turn every 40 seconds. () How long does it tke the sils to turn through: (i) 180 (ii) 90 (iii) 45 Wht ngle do the sils turn through in: (i) 30 seconds, (ii) 15 seconds, (iii) 25 seconds 10. The digrm shows the positions of Json nd din. The rrow shows the direction of north. din Json () (e) Copy or trce the digrm. Kren is west of din nd north of Json. Mrk Kren's position on your digrm. Jenny is est of Json nd southest of din. Mrk Jenny's position on your digrm. Wendy is west of Jenny nd southest of Kren. Where is Wendy in reltion to din Ji is north of Json nd south of Kren. Descrie where he could e stnding. 6
30 1.2 Mesuring Angles The ojectives of this section re to understnd how to use protrctor to mesure ngles e le to drw ngles, using protrctor understnd how to interpret dt shown in pie chrts e le to use pie chrts to illustrte dt. ote The ngle round complete circle is is 360. The ngle round point on stright line is 180. 360 o 180 o A right ngle is 90. 90 Exmple 1 Mesure the ngle CAB in the tringle shown. C A B Solution Plce protrctor on the tringle s shown. The ngle is mesured s 47. 50 C 40 20 A 10 0 B 7
30 1.2 Exmple 2 Mesure this ngle. Solution Using protrctor, the smller ngle is mesured s 100. 110 100 So required ngle = 360 100 = 260 90 80 70 60 Exmple 3 Drw ngles of () 120 330. 0 40 50 30 10 20 Solution () Drw horizontl line. Plce protrctor on top of the line nd drw mrk t 120. 120 110 100 90 80 70 60 50 40 20 10 0 Then remove the protrctor nd drw the ngle. 120 8
1.2 To drw the ngle of 330, first sutrct 330 from 360 : 360 330 = 30 Drw n ngle of 30. The lrger ngle will e 330. 30 30 330 330 Exercises 1. For ech of the following ngles, first estimte the size of the ngle nd then mesure the ngle to see how good your estimte ws. () (e) (f) 9
1.2 (g) (h) 2. Estimte nd mesure the size of ech of these reflex ngles. () 10
1.2 3. () Mesure ech of the ngles in this pie chrt. Other Fvourite UK footll tems for students in Grde 7 Chelse Mnchester United ewcstle Arsenl Explin how you cn tell tht Mnchester United is the most populr of these tems. Which is the second most populr tem 4. Drw the following ngles: () 20 42 80 105 (e) 170 (f) 200 (g) 275 (h) 305 5. In which of these polygons re the ngles ll the sme size Find ll the ngles in ech polygon. (You my need to copy the shpes on to pper nd extend the lines.) () B C B A C A D E D F E 11
1.2 B C B A D A C H E G D G F F E 6. () Drw the shpe elow, where O is the centre of the circle. Mke the rdius of your circle 6 cm. A C O 100 95 B Mesure the distnces etween AB, BC nd AC. 7. Rvinder finds out the fvourite sports for memers of his clss. He works out the ngles in the list shown opposite for pie chrt. Drw the pie chrt. Sport Angle Footll 110 Swimming 70 Tennis 80 Rugy 40 Hockey 30 Bdminton 10 Other 20 12
1.3 Clssifying Angles The ojectives of this section re to understnd how to identify cute, otuse nd reflex ngles e le to drw shpes to include the vrious ngles. Angles of less thn 90 re cute ngles Angles etween 90 nd 180 re otuse ngles Angles etween 180 nd 360 re reflex ngles So you cn esily identify the three types of ngle. Here re some exmples. Acute Otuse Acute Otuse Reflex Reflex Reflex Acute Exercises 1. Is ech of the following ngles cute, otuse or reflex () 13
1.3 (e) (f) 2. For ech shpe elow stte whether the ngle t ech corner is cute, otuse or reflex. () B C A D B F D C A E 3. () Drw tringle with one otuse ngle. Drw tringle with no otuse ngles. 4. Drw four-sided shpe with: () one reflex ngle, two otuse ngles. 1.4 Angles on Line nd Angles t Point The ojectives of this section re to understnd tht ngles on stright line dd up to 180 understnd tht ngles t point dd up to 360 clculte the size of unknown ngles using these two fcts. Rememer tht () ngles on line dd up to 180 180 14
1.4 nd ngles t point dd up to 360. 360 These re two importnt results which help when finding the size of unknown ngles. Exmple 1 Wht is size of the ngle mrked 45 55 Solution 45 + 55 = 100 So ngle = 180 100 = 80 Exmple 2 Wht is the size of the ngle mrked 150 70 Solution 70 + 150 = 220 So ngle = 360 220 = 140 Exercises 1. Clculte the unknown ngle in ech of the following digrms. () 70 85 15
1.4 55 112 60 80 (e) (f) 22 62 27 126 45 2. Clculte the unknown ngle in ech digrm. () 92 124 278 30 255 30 118 62 (e) (f) 21 57 41 80 30 60 110 16
1.4 3. Some of the ngles in the pie chrt hve een clculted: Red 90 Blue 95 Purple 50 BLUE PURPLE RED YELLOW Wht is the ngle for yellow 4. The picture shows tipper truck. 80 70 () Find the ngles mrked nd. The 80 ngle decreses to 75 s the tipper tips further. Wht hppens to ngle 5. The digrm shows plyground roundout viewed from ove. Five metl rs re fixed to the centre of the roundout s shown. The ngles etween the rs re ll the sme size. () Wht size re the ngles Wht size would the ngles e if there were 9 metl rs insted of 5 6. 142 A oy hngs punchg on wshing line. Find the unknown ngles if oth ngles re the sme size. 17
1.4 7. The digrm shows two stright lines. Find the ngles, nd c. Wht do you notice c 28 8. In the digrm the lrge ngle is 4 times igger thn the smller ngle. Find the two ngles. 9. The picture shown jck tht cn e used to lift up cr. Find the ngles mrked nd. 56 126 10. The digrm shows regulr hexgon. () Find the size of ech of the ngles mrked t the centres of the hexgon. Wht would these ngles e if the polygon ws decgon (10 sides). If the ngles were 30, how mny sides would the polygon hve 1.5 Finding Angles in Tringles The ojectives of this section re to understnd tht the interior ngles of tringle lwys sum to 180 recognise whether tringle is isosceles, equilterl or sclene find unknown interior nd exterior ngles of tringle. 18
1.5 The interior ngles of ny tringle will lwys sum (dd up) to 180. + + c = 180 exterior ngle interior ngles exterior ngle c exterior ngle Exmple Find the ngle mrked in the digrm opposite. Solution 70 + 50 = 120 So 180 120 = 60 nd = 60 70 50 The finl prt of this section dels with the clssifiction of tringles. ISOSCELES TRIAGLE Equl ngles Sides of equl length EQUILATERAL TRIAGLE All sides re the sme length All ngles re 60 SCALEE TRIAGLE All sides hve different lengths. All ngles re of different sizes. 19
1.5 Exercises 1. Find the unknown ngle in ech tringle. () 80 50 35 120 94 23 81 62 (e) 38 (f) 91 17 41 2. Find the unknown ngles in ech of the following tringles. () 80 8 cm 5 cm 5 cm 85 8 cm 122 3 cm 4 cm 4 cm 3 cm 4 cm c 20
1.5 3. Stte whether ech tringle elow is isosceles, equilterl or sclene. () 45 5 cm 7 cm 45 8 cm 60 20 80 60 4. For ech tringle elow, find the unknown interior ngle nd the mrked exterior ngle. () 62 42 63 71 75 18 120 21
1.5 5. Explin how to find the exterior ngle without hving to clculte n interior ngle. Find the exterior ngles mrked on these tringles. () 41 18 61 124 47 108 25 6. Find the totl of the 3 exterior ngles for this tringle. 61 65 c Do you think you will get the sme nswer for different tringles Explin your nswer. 7. For ech of the following tringles, drw in the exterior ngles nd find their totl. () 30 65 50 65 20 22
1.5 58 38 62 Comment on your results. 8. Find the unknown ngle or ngles mrked in ech of the following digrms. () 52 152 121 111 140 130 c d 130 (e) (f) c d 110 c 23
1.5 9. Prt of roof is mde out of 4 similr isosceles tringles. 40 Copy the digrm nd mrk the sides tht hve the sme lengths. On your digrm, write in the size of ll the mrked ngles. 10. () For this isosceles tringle, find the other two interior ngles. 10 Find the other ngles if the 10 increses to 20 nd then to 30. Wht do you think will hppen if the 10 is incresed to 40 11. One ngle of n isosceles tringle is 70. Wht re the other ngles (There is more thn one solution!) 24
1.6 Prllel nd Intersecting Lines The ojectives of this section re to understnd nd e le to pply the rules for clculting unknown ngles. When line intersects (or crosses) pir of prllel lines, there re some simple rules tht cn e used to clculte unknown ngles. The rrows on the lines indicte tht they re prllel. e f c d ( ) = nd c = d, nd e = f These re clled verticlly opposite ngles. ( ) = c nd = d These re clled corresponding ngles. = c These re clled lternte ngles. + e = 180, ecuse djcent ngles on stright line dd up to 180. These re clled supplementry ngles. ote lso, tht c + e = 180 (llied or supplementry ngles) Exmple 1 In the digrm opposite, find the unknown ngles if = 150. d e Solution To find : + = 180 (ngles on stright line, supplementry ngles) 150 + = 180 = 30 c 25
1.6 To find c: c = (verticlly opposite ngles or ngles on stright line) c = 30 To find d: d = (corresponding ngles) d = 150 To find e: e = c (corresponding ngles) e = 30 Exmple 2 Find the size of the unknown ngles in the prllelogrm shown in this digrm: 70 Solution To find : + 70 = 180 (llied or supplementry ngles) = 110 c To find : + = 180 (llied or supplementry ngles) + 110 = 180 To find c: = 70 c + 70 = 180 (llied or supplementry ngles) or or c = 110 ( ) c = 360 + + 70 (ngle sum of qudrilterl) = 360 250 = 110 c = (opposite ngles of prllelogrm re equl) 26
1.6 Exercises 1. Which ngles in the digrm re the sme size s: (), e f h g d c 2. Find the size of ech of the ngles mrked with letters in the digrms elow, giving resons for your nswers: () 40 c 70 c d 110 c d 99 3. Find the size of the three unknown ngles in the prllelogrm opposite: 65 c 4. One ngle in prllelogrm mesures 36. Wht is the size of ech of the other three ngles 5. One ngle in rhomus mesures 133. Wht is the size of ech of the other three ngles 6. Find the sizes of the unknown ngles mrked with letters in the digrm: 50 30 c d e 27
1.6 7. () In the digrm opposite, find the sizes of the ngles mrked in the tringle. Give resons for your nswers. Wht specil nme is given to the tringle in the digrm c 37 å 143 8. The digrm shows icycle frme. Find the sizes of the unknown ngles, nd c. 46 20 A c 62 72 9. BCDE is trpezium. () Find the sizes of ll the unknown ngles, giving resons B C for your nswers. Wht is the specil nme given to this type of trpezium 126 D E 1.7 Berings The ojectives of this section re to understnd wht is ment y 'ering' e le to clculte nd mesure erings. Berings re mesure of direction, with north tken s reference. If you re trvelling north, your ering is 000. If you wlk from O in the direction shown in the digrm, you re wlking on ering of 110. O 110 28
1.7 Berings re lwys mesured clockwise from north, nd re given s three figures, for exmple: 240 60 330 Bering 060 Bering 240 Bering 330 Exmple 1 On wht ering is ship siling if it is heding: W E () E, S, c) W, SE, (e) W W E Solution () SW S SE 90 180 E Bering is 090. Bering is 180. S W Bering is 270. 270 (e) W 135 315 SE Bering is 135 Bering is 315 29
1.7 Exmple 2 A ship sils from A to B on ering of 060. On wht ering must it sil if it is to return from B to A Solution The digrm shows the journey from A to B. Extending the line of the journey llows n ngle of 60 to e mrked t B. Bering of A from B = 60 + 180 = 240 nd this is clled ck ering or reciprocl ering. Exercises 60 A 1. Wht ngle do you turn through if you turn clockwise from: () to S, E to W, to E, to SW, (e) W to W 60 B 2. Copy nd complete the tle: Direction E W SW Bering 3. The mp of n islnd is shown elow: Mine Quy Church Tower Bech Lighthouse Wht is the ering from the tower, of ech plce shown on the mp 30
1.7 4. The digrm shows the positions of two ships, A nd B. () Wht is the ering of ship A from ship B Wht is the ering of ship B from ship A A 5. The digrm shows 3 plces, A, B nd C. Find the ering of: () A from C, B from A, C from B, B from C. A B B 6. In Cnd, n eroplne flies from Victori to Edmonton on ering of 044. On wht ering should the pilot fly, to return to Edmonton from Victori 7. On four seprte occsions, plne leves Kingston to fly to different destintion. The erings of these destintions from ormn Mnley Airport, Kingston re given elow. Destintion Bering Pris, Frnce 077 ssu, Bhms 356 C Sntigo de Cu 036 Crtgen, Colomi 162 Copy nd complete the digrm to show the direction in which the plne flies to ech destintion. Kingston Pris, Frnce 31
1.7 8. A ship sils W from port to tke supplies to n oil rig. On wht ering must it sil to return from the oil rig to the port 9. If A is north of B, C is southest of B nd on ering of 160 from A, find the ering of: () A from B, A from C, C from B, B from C. 10. If A is on ering of 300 from O, O is E of B, nd the ering of B from A is 210, find the ering of: () A from B, O from A, O from B. 1.8 Scle Drwings The ojectives of this section re to construct scle drwings, using erings use these scle drwings to solve prolems. Exmple 1 A ship sils 20 km E, then 18 km S, nd then stops. () How fr is it from its strting point when it stops On wht ering must it sil to return to its strting point Solution The pth of the ship cn e drwn using scle of 1 cm for every 2 km, s shown in the digrm. A 180 20 km 18 km O 45 Scle: 1 cm = 2 km B 32
1.8 () The distnce BO cn e mesured on the digrm s 7.3 cm which represents n ctul distnce of 14.6 km. The ering of O from B cn e mesured s 285. ote: Rememer to lwys put the scle on the digrm. Exmple 2 A mn wlks 750 m on ering of 030. He then wlks on ering of 315 until he is due north of his strting point, nd stops. () How fr does he wlk on the ering of 315 How fr is he from his strting point when he stops Solution A scle drwing cn e produced, using scle of 1 cm to 100 m. B A 315 750 m 30 O () The distnce AB cn e mesured s 5.4 cm, which represents n ctul distnce of 530 m. The distnce OB cn e mesured s 10.2 cm, representing n ctul distnce of 1020 m. 33
1.8 Exercises 1. A girl wlks 80 m north nd then 200 m est. () How fr is she from her strting position On wht ering should she wlk to get ck to her strting position 2. Andrew wlks 300 m W nd then wlks 500 m south nd then stops. () How fr is he from his strting position when he stops On wht ering could he hve wlked to go directly from his strting position to where he stopped 3. An eroplne flies 400 km on ering of 055 It then flies on ering of 300, until it is due north of its strting position. How fr is the eroplne from its strting position 4. A cptin wnts to sil his ship from port A to port B, ut the journey cnnot e mde directly. Port B is 50 km north of A. The ship sils 20 km on ering of 075. It then sils 20 km on ering of 335 nd then drops nchor. () How fr is the ship from port B when it drops nchor On wht ering should the cptin sil the ship to rrive t port B Lnd B Se A 5. Juli intended to wlk 200 m on ering of 240. Her compss did not work properly, so she ctully wlked 200 m on ering of 225. Wht distnce nd on wht ering should she wlk to get to the plce she intended to rech 6. A hot ir lloon is lown 5 km W. The wind then chnges direction nd the lloon is lown further 6 km on ering of 300 efore lnding. How fr is the lloon from its strting point when it lnds 7. Ronldo nd Jde set off wlking t the sme time. When they strt, Ronldo is 6 km W of Jde. Jde wlks 3 km on ering of 350 nd Ronldo wlks 4 km on ering of 020. How fr prt re they now 34
1.8 8. An eroplne flies 200 km on ering of 335. It then flies 100 km on ering of 170 nd 400 km on 280, nd then lnds. () How fr is the eroplne from its strting point when it lnds On wht ering could it hve flown to complete its journey directly 9. Billy is siling on ering of 135. After his ot hs trvelled 20 km, he relises tht he is 1 km north of the port tht he wnted to rech. () On wht ering should he hve siled How fr from his strting point is the port tht he wnted to rech 10. A pilot knows tht to fly to nother irport he needs to fly 500 km on ering of 200. When he hs flown 400 km, he relises tht he is 150 km from the irport. () On wht ering hs the pilot een flying On wht ering should he fly to rech the irport (ote tht there re two nswers.) 11. A Jmicn cellphone compny needs to clculte the erings nd distnces from its phone mst in Ewrton to severl other towns nd villges on the islnd. The erings nd the distnces from Ewrton to these loctions re given in the tle elow. Loction Bering Distnce Orcess 030 30 km Runwy By 318 40 km Hyes 208 36.5km Linsted 127 6.5 km Using scle of 1 cm to represent 5 km, drw mp showing the positions of the five loctions. 35