Plane figure geometry
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- Kristopher Sparks
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1 2 lne figure geometry ontents: E F G H I Turning Mesuring ngles lssifying nd nming ngles omplementry nd supplementry ngles ngles in revolution isecting ngles onstructing 9 ngles to line lne shpes oints nd lines
2 34 LNE FIGURE GEOMETRY (hpter 2) If we look crefully we notice ngles in mny objects nd situtions. For exmple, we see ngles in building frmeworks, pitches in roof structures, steepness of rmps, nd the positioning of bots nd eroplnes from their home bses. ngle mesurement dtes bck more thn 25 yers nd is still very importnt tody in rchitecture, building, surveying, engineering, nvigtion, spce reserch, etc. EGREE MESURE To mesure the size of ngles ccurtely we use smll unit of mesure. This unit is clled degree. There re 36 degrees, 36 o, in one complete turn or revolution. egrees hve been used from ncient bylonin times when ll the numbers were written nd clculted in bse o is useful mesure becuse it is divisible by 2, 3, 4, 5, 6, 8, 9,, 12, etc., nd this mkes it esy to find frctions of 36 o such s hlves, thirds, qurters, nd so on. OENING ROLEM There re four posts t the corners,, nd of pddock with rised mound s shown. The distnces between the posts re esily mesured nd re shown on the figure. 113 m Your tsk is to drw n ccurte scle digrm of the sitution. ecuse of the mound you cnnot ccurtely mesure the distnces or. 84 m Things to discuss, think bout nd do: rough figure ² Is knowing the lengths of the four sides enough to ccurtely redrw the figure? Try doing this using millimetres insted of metres. ² Wht other fcts re necessry to ccurtely redrw the figure? ² How would you ccurtely mesure the four ngles t,, nd in the rel world? ² How would you ccurtely mesure the four ngles of the figure on pper? ² If the ngle t is known to be o, could you ccurtely peg out on the school ovl 1 th scle model of pddock? You my only use four pegs, lrge mesuring tpe (s used on thletics dys) nd n ngle mesuring device (such s theodolite or modern lser equipment). 125 m 97 m
3 LNE FIGURE GEOMETRY (hpter 2) 35 TURNING Mny mchines re operted with dil showing different settings. The dil is rotted in order for the mchine to perform prticulr opertion. Exmple 1 Self Tutor Here is the dil of ceiling fn. nticlockwise There re three settings s well s n off position, s shown. i Wht setting will be beside the rrow for ech of the following rottions? ii Through how mny degrees hs the dil turned in ech cse? 1 4 turn clockwise b 1 2 c 3 4 turn clockwise turn nticlockwise d full revolution nticlockwise i low ii 1 4 of 36o =9 o b i medium ii 1 2 of 36o = 1 o c i low ii 3 4 of 36o = 2 o d i off ii 36 o clockwise EXERISE 2 1 Using the fct tht full rottion or turn is equl to 36 o, wht is the size of: hlf rottion b qurter turn c three-qurter turn d three full turns e one nd hlf rottions f one-eighth rottion? 2 For the ceiling fn described in the exmple bove, copy ech digrm nd fill in the missing settings: b 3 n industril fn is used to remove chemicl fumes from fctory. It hs five settings s shown, nd cn only be turned clockwise. b There re six equl divisions on the dil. How mny degrees is ech division worth? How mny degrees does ech of the following rottions require? i off to medium ii off to very high iii high to off VERY HIGH HIGH OFF ME clockwise VERY LOW LOW
4 LNE FIGURE GEOMETRY (hpter 2) MESURING NGLES n ngle is mde up of two rms which meet t point clled the vertex. The size of the ngle is mesured by using the mount of turning (rottion) from one rm to the other. MESURING EVIES We use protrctor (or geoliner) which hs tiny 1 o mrkings on it vertex 9 1 ngle rm rm centre To use protrctor to mesure ngles we: ² plce it so its centre is t the ngle s vertex nd o lies exctly on one rm ² strt t o nd follow the direction the ngle turns through to rech the other rm. bse line The smll symbol mens degrees. Exmple 2 Self Tutor Mesure these ngles: b This ngle hs size 47 o. b This ngle hs size 123 o :
5 1 Wht is the size of the ngle being mesured? b c d e f g h EXERISE 2 i j LNE FIGURE GEOMETRY (hpter 2) 37
6 38 LNE FIGURE GEOMETRY (hpter 2) 2 Look t the following ngles nd estimte their sizes. Estimte Mesure Record your estimtes. Now mesure ech ngle ccurtely with protrctor nd record your nswers. opy nd complete tble like the one given. How close b ws your estimte to the mesured ngle?. b c d e LSSIFYING N NMING NGLES ngles re clssified ccording to their size. Revolution Stright ngle Right ngle One complete turn. 1 2 turn. 1 4 turn. One revolution = 36 o : 1 stright ngle = 1 o : 1 right ngle =9 o : cute ngle Obtuse ngle Reflex ngle Less thn 1 4 turn. etween 1 4 turn nd 1 2 turn. etween 1 2 turn nd 1 turn. n cute ngle hs size n obtuse ngle hs size reflex ngle hs size between o nd 9 o : between 9 o nd 1 o : between 1 o nd 36 o : NMING NGLES To void confusion in complicted figures, three point ngle nottion is used. The shded ngle is nmed ] or ] where the vertex must be the middle letter. lling the shded ngle unstisfctory. Why? ], would be
7 LNE FIGURE GEOMETRY (hpter 2) 39 EXERISE 2 1 Mtch the nmes to the correct ngles: ] b ] c ] d ] 2 rw nd lbel digrm for ech of the following ngles: ]EF b ]ZXY c ]XYZ d ]QR e ]RQ 3 Using only ruler nd pencil, drw ngles you estimte to be: 9 o b 45 o c o d 6 o e 135 o heck your estimtions using protrctor. 4 Find the size of ech of the following ngles without your protrctor: ] b ] c ] d ]E E Use your ruler nd protrctor to drw ngles with the following sizes: 35 o b 131 o c 258 o Get someone else to check the ccurcy of your ngles. 6 Which is the lrger ngle, ] or ]XYZ? Y Z 7 rw free-hnd sketch of: X cute ngle Q b right ngle NXZ c stright ngle QT d obtuse ngle T e reflex ngle SM f revolution ]E 8 Use protrctor to mesure the nmed ngles: i ]MN ii ]OL b i ]VTU ii reflex ]VST iii ]ON iii reflex ]TVU S Q L M T N O V U
8 4 LNE FIGURE GEOMETRY (hpter 2) 9 Kim hits the billird bll so tht it follows the pth shown. Wht cute ngle will it mke with the edge of the tble? golfer completing his swing holds the golfclub behind his body. Wht is the size of the reflex ngle between his body nd the club? fn switch hs seven operting speeds prt from the off position (). If the knob is turned clockwise to 5, through how mny degrees hs it rotted? b How mny less degrees would it hve to turn if it rotted nticlockwise? 12 How mny obtuse ngles cn be drwn joining ny three dots? b How mny right ngles cn be drwn? OMLEMENTRY N SULEMENTRY NGLES Two ngles re complementry if their sizes dd to 9 o. Two ngles re supplementry if their sizes dd to 1 o. For exmple, O M U N T ]MNO nd ]MN re complementry ]STU nd ]UTV re supplementry becuse 52 o +38 o =9 o. becuse 79 o + 1 o = 1 o. S V
9 LNE FIGURE GEOMETRY (hpter 2) 41 Note: Two ngles re equl if they hve the sme size (degree mesure). Exmple 3 Self Tutor re ngles with sizes 37 o nd 53 o complementry? b Wht ngle size is the supplement of 48 o? 37 o +53 o =9 o. So, the ngles re complementry. b The ngle size is 1 o 48 o = 132 o. EXERISE 2 1 dd the following pirs of ngles nd stte whether they re complementry, supplementry, or neither: 2 o, o b o, o c 1 o, 4 o d 47 o, 43 o e 7 o, 63 o f 35 o, 55 o 2 Find the size of the ngle complementry to: o b 5 o c 85 o 3 Find the size of the ngle supplementry to: o b 5 o c 9 o 4 lssify the following ngle pirs s complementry, supplementry or neither: ]O nd ]O b ]O nd ]OE The symbol c ]O nd ]OE indictes d ]O nd ]OE right ngle E O 5 opy nd complete: the size of the complement of x o is... b the size of the supplement of y o is... Exmple 4 Self Tutor Find the vlue of the pronumerl in: b x x 78 The ngles 58 o nd x o re complementry. ) x =9 58 ) x =32 b The three ngles dd up to 1 o. ) x = ) x =
10 42 LNE FIGURE GEOMETRY (hpter 2) 6 Find the vlue of the pronumerl in: b c b c c 62 d e f 12 e d 33 g h i f f 38 g 32 g 48 h h 74 x x x E The ngles round point totl 36 o. NGLES IN REVOLUTION So, ] + ] + ] = 36 o. Exmple 5 Self Tutor Find in: The sum of the five ngles is 36 o. ) the three equl ngles dd to 36 o 9 o o = o So, ech must be o 3=5 o ) =5: EXERISE 2E 1 Find the size of the ngles mrked with pronumerl: b c 266 y z 27 x
11 2 Find the vlue of the pronumerls in: b c p 35 r r q 6 d e f LNE FIGURE GEOMETRY (hpter 2) s t t u u w F ISETING NGLES When we bisect n ngle with stright line we divide it into two ngles of equl size. The following construction shows ngle bisection with compss nd ruler only. Exmple 6 isect ngle. Self Tutor Step 1: With centre, drw n rc of convenient rdius which cuts nd t nd Q respectively. Q Step 2: With Q s centre, drw n rc within the ngle. Q Step 3: Keeping the sme rdius nd with centre drw nother rc to intersect the previous one t M. Q M Step 4: Join to M. M bisects ngle, so ]M = ]M. M Q
12 44 LNE FIGURE GEOMETRY (hpter 2) EXERISE 2F 1 Use your protrctor to drw n ngle of size o : isect ngle, using compss nd ruler only. b heck with your protrctor the size of ech of the two ngles you constructed. 2 rw n cute ngle XYZ of your own choice. isect the ngle without using protrctor. b heck your construction using your protrctor. 3 rw n obtuse ngle of your own choice. isect the ngle using compss only. b heck your construction using your protrctor. 4 Using ruler, drw ny tringle with sides greter thn 5 cm. b isect ech ngle using compss nd stright edge only. c Wht do you notice bout the three ngle bisectors? G ONSTRUTING 9 NGLES TO LINE right ngle or 9 o ngle cn be constructed without protrctor or set squre. onsider the following exmple: Exmple 7 Self Tutor onstruct n ngle of 9 o t on the line segment XY. X Y Step 1: On line segment XY, drw semicircle with centre nd convenient rdius which cuts XY t M nd N. X M N Y Step 2: With centre M nd convenient rdius lrger thn M, drw n rc bove. X M N Y W Step 3: With centre N nd sme rdius drw n rc to cut the first one t W. X M N Y
13 LNE FIGURE GEOMETRY (hpter 2) 45 Step 4: rw the line from through W. ngles WY nd WX re 9 o. W X M N Y EXERISE 2G 1 rw horizontl line segment which is 5 cm long. b Use your ruler to find on where =2:5 cm. c onstruct 9 o ngle t using your compss nd ruler. Horizontl lines re drwn cross the pge like horizon. 2 rw line segment of length 3 cm. b onstruct n ngle of 9 o t. (You will need to extend the line segment on the other side of.) c rw of length 4 cm, such tht ] is 9 o. d Join to form tringle nd mesure the length of. 3 rw line segment XY of length 8 cm. b t X construct n ngle of 9 o. c rw XZ of length 5 cm. Join ZY. d Use your compss to bisect ngle ZXY. e Wht is the size of ngle YX? f If ZY nd X meet t Q, mesure QY to the nerest mm. Z 5cm X Q 8cm Y H LNE SHES shpe tht is drwn on flt surfce or plne is clled plne figure. If the shpe hs no beginning or end it is sid to be closed. We cn use properties like these to clssify objects. OLYGONS olygons re closed plne figures with stright line sides which do not cross over themselves. Simple exmples re: tringles qudrilterls pentgons
14 46 LNE FIGURE GEOMETRY (hpter 2) Exmple 8 Self Tutor lssify s polygon or not, giving resons: b c b c It is closed nd hs stright line sides, so it is polygon. It does not hve ll stright line sides, so it is not polygon. It is not closed nd so it is not polygon. olygons re nmed ccording to the number of sides they hve. For exmple, 9 sided polygon cn be clled 9-gon. However, mny polygons re known by other more fmilir nmes. Here re the first few: Number of Sides olygon Nme 3 Tringle 4 Qudrilterl 5 entgon 6 Hexgon 7 Heptgon 8 Octgon 9 Nongon ecgon IRLES Suppose we mke loop from length of light rope nd plce it over fixed spike in the ground. The rope is mde tut nd stick is plced t the opposite end to the fixed spike. y keeping the rope tut nd moving the stick round the spike, circle is produced. The fixed spike is the circle s centre. This method of drwing circle ws known nd used by builders in ncient Egypt. rts of circle this is circle this is the circle s centre (nd is often not drwn) this is rdius which joins the circle s centre to ny point on the circle is dimeter of the circle s it psses through the circle s centre nd its end points lie on the circle
15 LNE FIGURE GEOMETRY (hpter 2) 47 this is semi-circle (hlf of circle) this is n rc of the circle (prt of the circle) this is chord of the circle this is segment of the circle this is sector of the circle EXERISE 2H 1 lssify s polygons or not, giving resons: b c d convex (no reflex ngles) e f g h non-convex (hs reflex ngle) 2 rw free-hnd sketch of: convex 4-sided polygon b non-convex 4-sided polygon c convex 5-sided polygon d non-convex 6-sided polygon 3 Wht specil nme is given to polygon with: three sides b four sides c five sides d six sides e eight sides f ten sides 4 Mtch the prt of the figure indicted to the phrse which best describes it: b c d TIVITY e f g h i j E F G H I J semi-circle rdius minor rc mjor rc dimeter chord minor segment mjor segment mjor sector minor sector
16 48 LNE FIGURE GEOMETRY (hpter 2) GROU TIVITY GEOMETRIL IES Wht to do: 1 In the clssroom find two exmples of ech of the following: ² point ² line ² n ngle ² flt surfce ² curved surfce 2 iscuss the following: ² Wht is ment by point? ² How smll cn point be? I OINTS N LINES OINTS IN GEOMETRY Exmples of point in the clssroom re: ² the intersection of two djcent wlls nd the floor ² speck of dust in the room t prticulr instnt in time. In geometry, point could be represented by smll dot. To help identify it we could nme it with or ssign cpitl letter. onsider: The letters, nd re useful for identifying points to which we re referring. We cn mke sttements like: the distnce from to is... or the ngle t mesures..., etc. To mthemticin, point mrks position nd does not hve ny size. FIGURES N VERTIES ny specil collection of points in plne (flt surfce) is clled figure. The figure longside contins four points which hve been lbelled,, nd. These corner points re lso known s vertices. ( Note: oint is vertex of the figure.) Vertices is the plurl of vertex.
17 LNE FIGURE GEOMETRY (hpter 2) 49 GROU ISUSSION LINES Wht to do: iscuss the following questions: 1 How mny different stright lines could be drwn through single point? 2 Suppose nd re two different points. How mny stright lines cn be drwn such tht ech line psses through both nd? STRIGHT LINES stright line (usully clled line) is continuous infinite collection of points which lie in prticulr direction. This line which psses through points nd is clled line or line. This prt line which joins points nd is clled line segment or line segment. Note: ² When three or more points lie on line we sy tht the points re colliner. The points,, nd shown re colliner. ² If three or more lines meet (intersect) t the sme point we sy tht the lines re concurrent. The lines shown re concurrent t point. RLLEL N INTERSETING LINES Imgine plne, like tble top, which goes on indefinitely in ll directions, i.e., it hs no boundries. In such plne, two stright lines re either prllel or intersecting. rrow heds re used to show prllel lines. prllel point of intersection rllel lines re lines which re lwys fixed distnce prt nd so never meet. EXERISE 2I 1 Give two exmples in the clssroom of: point b line c flt surfce 2 escribe with sketch the mening of: vertex b n ngle c point of intersection d prllel lines e colliner points f concurrent lines
18 5 LNE FIGURE GEOMETRY (hpter 2) 3 Give ll wys of nming the following lines: b Q (6 nswers) 4 Nme the point of intersection of: line 1 nd line 2 b line 2 nd line 3 c line nd line MN M line 2 line 3 line 1 N 5 is qudrilterl nd line segment is clled digonl. Nme the four sides of the qudrilterl. b Nme the two digonls of the qudrilterl. c t wht point do the digonls meet? d How mny line segments meet t? e Wht cn be sid bout points, X nd? f Wht cn be sid bout the line segments, nd? 6 How mny different lines do you think you cn drw through: two points nd b ll three colliner points, nd c one point d ll three non-colliner points,,? 7 rw different digrm to fit ech sttement: is point on line. b Lines nd meet t point X. c oint does not lie on line. d X, Y nd Z re colliner. e Line segments, nd EF re concurrent t G. X 8 Nme line in two other wys. b Nme two different lines contining point. c Wht cn be sid bout i points, nd ii lines F nd iii lines FE, nd? F E 9 When drwing lines through three different points there re two possible cses: 3 different lines 1 line, s the points re colliner b c How mny different cses cn we hve for four different points? Illustrte ech cse. Now drw the cses for five different points. You my lso wish to experiment with higher number of points.
19 LNE FIGURE GEOMETRY (hpter 2) 51 REVIEW SET 2 1 Stte the complement of 41 o. b Wht is the size of one revolution? c etermine the mesure of ngle. 2 For the given figure: find the ngle number corresponding to: i ] ii ] iii ] b clssify the following ngles s cute, obtuse or reflex: i 3 ii 1 iii Find the vlue of the pronumerl in: b c n x x y 14 4 Nme the following polygons: b 5 efine the following terms: n rc of circle b sector of circle 6 If the rdius of circle is 5 cm in length, wht is the length of dimeter? 7 How mny points re needed to determine the position of line? 8 rw digrm to illustrte the following sttement: Line segments nd intersect t. 9 Nme line RS in two other wys. R b Nme two different lines contining Q point. S c Wht cn be sid bout U i points, Q nd R T ii lines Q nd RS? Use your protrctor to drw n ngle of 56 o. isect this ngle using your compss. heck tht the two ngles produced re 28 o ech.
20 52 LNE FIGURE GEOMETRY (hpter 2) REVIEW SET 2 1 rw digrm to illustrte n obtuse ngle. b If ngle nd ngle EF re equl ngles nd the mesure of ngle is 72 o, find the mesure of ngle EF. c etermine the mesure of the reflex ngle QR. 2 Mtch the ngle nme to the ngle number: 1 6 ] b ] c ] Mtch ech ngle description to one of the ngles shown: revolution b right ngle c stright ngle d reflex ngle e cute ngle f obtuse ngle Q R E F 4 Wht is the complement of 63 o? b Wht is the supplement of o? 5 Find, giving reson, the vlue of y in ech of the following: b c 123 y y 131 y y y 6 Nme the following polygons: b 7 rw circle with rdius 2 cm. Mrk nd lbel: rdius b sector c segment d minor rc 8 opy nd complete: oints re colliner if Stte the totl number of points of intersection of the lines: rw of length 4 cm. onstruct n ngle of 9 o t using compss nd ruler only, nd drw of length 3 cm nd perpendiculr to. rw in nd mesure the length of.
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