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1 mesurement nd geometry topic 5 Geometry 5.1 Overview Why lern this? Geometry llows us to explore our world in very preise wy. uilders, rhitets, surveyors nd engineers use knowledge of geometry to ensure uildings re stle nd visully plesing. lot of geometry is lso involved in sport. Creful estimtion of ngles is required when plying pool or shooting for gol in hokey nd footll. Wht do you know? 1 think List wht you know out ngles, tringles nd qudrilterls. Use thinking tool suh s onept mp to show your list. 2 PIr Shre wht you know with prtner nd then with smll group. 3 shre s lss, rete thinking tool suh s lrge onept mp tht shows your lss s knowledge of ngles, tringles nd qudrilterls. Lerning sequene 5.1 Overview 5.2 Mesuring ngles 5.3 Construting ngles with protrtor 5.4 Types of ngles nd nming ngles 5.5 Tringles 5.6 Qudrilterls nd their properties 5.7 Prllel nd perpendiulr lines 5.8 Review ONLINE ONLY 144 Mths Quest 7

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3 mesurement nd geometry Digitl dos SkillSHEET Reding the sle on protrtor do-6447 SkillSHEET Compring the size of ngles do-6448 SkillSHEET Estimting the size of n ngle do-6449 SkillSHEET Mesuring ngles with protrtor do Mesuring ngles n ngle is mde when two lines meet t point. The point where the two lines meet is lled vertex nd the lines re lled rms of the ngle. Vertex rms ngles re mesured in degrees. One full revolution is equl to 360 degrees. The symol for degrees is. The devie for mesuring ngles is lled protrtor. There re two types of protrtors: irulr nd semiirulr. The irulr protrtor Cirulr protrtors re mrked from 0 to To mesure ngles using irulr protrtor, follow these steps. Step 1: Ple the entre of the protrtor on the vertex of the ngle. Step 2: Mth the 0 horizontl entre line with one of the rms of the ngle. Step 3: Red the size of the ngle indited y the other rm. 146 Mths Quest 7

4 mesurement nd geometry WorKeD exmple 1 First estimte nd then mesure the size of eh of the following ngles using irulr protrtor. Comment on the ury of the estimte. think 1 Estimte the size of the ngle. Write the estimte. Note: The ngle is etween 0 nd 90 ut loser to Ple the entre of the protrtor on the vertex of the ngle. 3 Mth the horizontl line tht psses through the entre of the protrtor nd points to 0 with one of the rms of the ngle. 4 Red the size of the ngle indited y the other rm. (You my need to extend the line so tht you n red it on the protrtor.) 5 Comment on the ury of the estimte. 1 Estimte the size of the ngle. Write the estimte. Note: The ngle ppers to e pproximtely hlfwy etween 0 nd Repet steps 2 to 5 from prt. WrIte The estimte is pproximtely The size of this ngle is 60. The estimte ws n urte one. The estimte is pproximtely The size of this ngle is 42. The estimte ws resonle one. Topi 5 Geometry 147

5 mesurement ND geometry 1 Estimte the size of the ngle. Write the estimte. Note: The ngle is etween 90 nd 180 ut loser to Repet steps 2 to 5 from prt. The estimte is pproximtely 115. The semiirulr protrtor Semiirulr protrtors hve two sles: inner nd outer. Eh sle is mrked from 0 to 180. Either of the two sles n e used for mesuring ngles. To mesure ngles tht re less thn 180, follow these steps. Step 1: Line up the entre of the protrtor s seline with the vertex (tht is, where the vertil 90 line intersets with the horizontl 180 seline). Step 2: Mth the se of the protrtor with one of the rms of the ngle. Step 3: Use the sle on the se strting t 0 to red the numer of degrees spnned y the ngle. To mesure ngles whih re greter thn 180, follow these steps. Step 1: Mesure the ngle enlosed y the stright lines (tht is, the one tht is smller thn 180 ). Step 2: Sutrt the numer otined in step (1) from Outside sle The size of this ngle is 120. The estimte ws resonle one Inside sle Mths Quest 7

6 mesurement nd geometry WorKeD exmple 2 First estimte nd then mesure the size of the ngle shown t right using semiirulr protrtor. Comment on the ury of the estimte. think 1 Estimte the size of the ngle. Write the estimte. Note: The ngle ppers to e pproximtely hlfwy etween 0 nd Line up the entre of the protrtor s seline with the vertex (tht is, where the vertil 90 line intersets with the horizontl 180 seline). 3 Mth the se of the protrtor with one of the rms of the ngle. (You my need to extend the line so tht you n red it on the protrtor.) 4 Use the sle on the se strting t 0 (in this se n inner sle) to red the numer of degrees spnned y the ngle. 5 Comment on the ury of the estimte. WorKeD exmple 3 WrIte First estimte nd then mesure the size of the ngle shown t right using semiirulr protrtor. Comment on the ury of the estimte. think The estimte is pproximtely The size of this ngle is 45. The estimte ws n urte one. WrIte 1 Estimte the size of the ngle. Write the estimte. Note: The ngle ppers to e pproximtely hlfwy etween 270 nd 360. The estimte is pproximtely Topi 5 Geometry 149

7 mesurement nd geometry reflection Where in everydy life my I need to use my skills of mesuring ngles? Intertivity Digitl protrtor int Mesure the ngle enlosed y the stright lines. 3 Sutrt 50 from 360 to find the required ngle. 4 Comment on the ury of the estimte. Exerise 5.2 Mesuring ngles InDIVIDuL PtHWys PrCtIse Questions: 1, 2, 3, 4, 5, 6, 7 ConsoLIDte Questions: 1, 2, 3, 4, 5, 6, 7, 8 Individul pthwy intertivity int-4331 The smller ngle is 50. mster Questions: 1, 2, 3, 4, 5, 6, 7, 8 FLuenCy 1 WE1, 2, 3 First estimte nd then mesure the size of eh of the following ngles using protrtor. Comment on the ury of the estimte. d e f g h i j k l Required ngle: = 310 The estimte ws resonle one. 2 Comment on the ury of your mesurement ompred with the nswer in the text. 150 Mths Quest 7

8 mesurement nd geometry 3 MC The size of the ngle shown is losest to: C 40 D 150 e 140 The size of the ngle shown is losest to: C 142 D 145 e 38 understnding 4 In eh of the photogrphs, mesure the indited ngle. e 5 The reommended slope for wheelhir ess is 4 from the horizontl. n ngle greter thn this mkes it more diffiult for the person in the wheelhir to trvel up the rmp. d f (ontinued ) Topi 5 Geometry 151

9 mesurement ND geometry Would you rte the slope of the ottom setion of the rmp s diffiult, s reommended or esy? Why? How would you rte the seond prt of the rmp? RESONING 6 Study the rtoon of the golfer to help you nswer the following questions Copy the following tle. Digrm Estimte of ngle Mesured ngle Differene Without using protrtor, estimte the size of eh ngle in the first four digrms of the golf sequene ove nd omplete the seond olumn of the tle. Explin how you otined n estimte. Wht steps did you follow? d Complete the third olumn y mesuring eh ngle with protrtor. e Clulte the differene etween your estimte nd the tul vlue of eh ngle. f Desrie how you ould improve your skills in estimting the size of n ngle. g Try these improvements for estimting the size of the ngles in the remining four digrms. Copy nd omplete the following tle. Digrm Estimte of ngle Mesured ngle Differene h Hs the differene etween the estimted nd tul vlue for eh ngle eome smller? Hve your estimting skills improved? 152 Mths Quest 7

10 mesurement nd geometry ProLem solving 7 Using protrtor, mesure the ngles etween the two lok hnds shown in the digrms. Find wy to lulte the ngle etween the two rms without using protrtor. 8 Sen is flying kite. Depending on the position of the kite, the ngle etween the string nd the horizontl line hnges. C Fill in the following sttements: s the kite moves from point to point, the ngle inreses from to. s the kite moves from point to point C, the ngle from to. 5.3 Construting ngles with protrtor Protrtors n e used not only for mesuring, ut lso for onstruting ngles. To onstrut n ngle using irulr protrtor, follow the following steps. Step 1: Drw stright line. Step 2: Put dot t one end of the line. (The line now represents one of the rms nd the dot represents the vertex of the ngle.) Step 3: Ple the protrtor so tht its entre is on the vertex nd 0 is on the seline. Step 4: Find the required ngle on the sle nd mrk smll dot t the edge of the protrtor. Step 5: Join the smll dot with the vertex to form the seond rm of the ngle. Step 6: Lel the ngle. If the ngle to e onstruted is etween 0 nd 180, semiirulr protrtor n e used in extly the sme wy s the irulr one. Digitl do SkillSHEET Construting ngles with protrtor do-6451 Topi 5 Geometry 153

11 mesurement nd geometry WorKeD exmple 4 Construt eh of the following ngles think 1 Drw seline. Put dot t one end. This is to e the vertex of the ngle. WrIte/DrW 2 Ple the protrtor so tht its entre is on the vertex nd 0 is on the seline Find 50 on the sle nd mrk smll dot t the edge of the protrtor. 4 Join the smll dot nd the vertex with stright line. 5 Lel the ngle. 1 Repet steps 1 nd 2 from prt. 2 Find on the sle nd mrk smll dot t the edge of the protrtor. Rememer to strt t 0 nd, using the inside sle, move in n ntilokwise diretion. 3 Join the vertex nd the dot where you hve mrked the with stright line Lel the ngle. To onstrut n ngle greter thn 180 using semiirulr protrtor, follow the following steps. Step 1: Sutrt the given ngle from 360. Step 2: Construt the resulting ngle (whih will e less thn 180 ). Step 3: Lel the required ngle (it is outside the one tht hs een onstruted) Mths Quest 7

12 mesurement nd geometry WorKeD exmple 5 Construt n ngle of 210 using semiirulr protrtor. think It is not neessry to ple the protrtor upside down. The digrm t right shows the ngle of 210 (the sme s in Worked exmple 5), onstruted y pling the protrtor in the upright position. Exerise 5.3 Construting ngles with protrtor InDIVIDuL PtHWys PrCtIse Questions: 1, 2, 3, 4, 5, 6, 8 ConsoLIDte Questions: 1, 2, 3, 4, 5, 6, 7, 8 WrIte/DrW 1 Sutrt 210 from = Drw seline. Put dot t one end. This will eome the vertex of the ngle. 3 Ple the protrtor upside down on the seline so tht its entre is on the vertex. 4 Strt t 0 nd, using the outside sle, move in lokwise diretion; mrk the position of 150 with smll dot. 5 Remove the protrtor nd join the vertex to the smll dot representing Mrk the outside ngle s 210. Individul pthwy intertivity int-4332 mster Questions: 1, 2, 3, 4, 5, 6, 7, 8, 9 FLuenCy 1 WE4 Construt eh of the following ngles d 75.5 e 45 f 40.8 g 88 h 76 2 WE4 Construt eh of the following ngles d 143 e 140 f g 127 h WE5 Construt eh of the following ngles using semiirulr protrtor d 260 e 243 f 192 g h MC To onstrut n ngle of 212 using semiirulr protrtor, we need to onstrut n ngle less thn 180 first nd then lel the outside ngle. The size of the first ngle to e onstruted is: C 148 D 112 e none of these reflection In wht situtions would semiirulr protrtor e esier to use thn irulr one? Topi 5 Geometry 155

13 mesurement ND geometry 5 Construt eh of the following ngles d 349 e 328 f 300 g 345 h 358 UNDERSTNDING 6 Construt 20 ngle. On one side of this ngle drw 45 ngle. On the seond rm of the ngle in prt drw 108 ngle. d Wht is the remining reflex ngle? e Wht is the totl sum of the ngles? 7 Windmills use wind power to produe eletriity, pump wter nd mill grin. The three types of mhine shown in the digrm hve ldes or sils onstruted t equl ngles. Using protrtor nd ruler first mesure the ngles etween ldes nd then drw the three types of ldes. Wind turine ldes Windmills RESONING 8 Explin the method you would use to onstrut reflex ngle if you only hd 180 protrtor. 9 We see the ojets surrounding us euse light reflets of the surfe of the ojets s shown in the digrm. The two ngles shown re lwys equl. Imgine you re in room lit y light gloe s shown in the following digrm. Using protrtor nd ruler drw the next four pths followed y the ry of light shown in the digrm. Drw nother ry of light in different diretion. Use the sme proedure s in prt to drw the first four pths of the line drwn in prt. PROLEM solving 10 On n 4 sheet of pper drw 20 m long stright horizontl line lose to the ottom of the pge. Lel this line. Using protrtor onstrut n ngle of 30 t point nd n ngle of 30 t point. Extend the two lines until they interset. Cut out the tringle nd ompre it with the tringles onstruted in this wy y other students. Write sttement out your findings. 11 Drw ny tringle on n 4 sheet of pper. Use protrtor to mesure the three ngles. Cut out the tringle nd then ut the ngles s shown in the digrm. Ple the ngles next to eh other so tht their rms touh (see digrm). d Mesure the new ngle formed. C e Compre your results with other students nd write C sttement to summrise your findings. Sil 156 Mths Quest 7

14 mesurement nd geometry 5.4 Types of ngles nd nming ngles Types of ngles ngles n e lssified ording to their size. n ute ngle is greter thn 0, ut less thn 90. right ngle is n ngle tht equls extly 90. n otuse ngle is greter thn 90 ut less thn 180. stright ngle equls extly 180. reflex ngle is greter thn 180 ut less thn 360. revolution or perigon is n ngle of 360 ( full irle). Two ngles tht dd to give 90 re lled omplementry ngles. Eh ngle is the omplement of the other. Two ngles tht dd to give 180 re lled supplementry ngles. Eh ngle is the supplement of the other. WorKeD exmple 6 Clssify eh of the following ngles ording to its size think The given ngle is lrger thn 90, ut smller thn 180, so lssify it ordingly. The given ngle is etween 0 nd 90, so lssify it ordingly. The given ngle is lrger thn 180, ut less thn 360, so lssify it ordingly. WrIte Nming ngles ngles n e nmed using pitl letters of the English lphet. ommon wy to nme n ngle is to use three letters: two letters to represent the rms of the ngle, nd third letter to represent its vertex. The letter representing the vertex is lwys pled in the middle (etween the two letters representing the rms). Insted of writing the word ngle, we use the symol. 115 is n otuse ngle. 27 is n ute ngle. 300 is reflex ngle. Digitl dos SkillSHEET Clssifying ngles do-6452 SkillSHEET Nming ngles do-6453 Topi 5 Geometry 157

15 mesurement nd geometry WorKeD exmple 7 Nme eh of the following ngles. X reflection Why is it importnt to lwys put the letter representing the vertex in the middle of the ngle s nme? O think Y Nme the ngle y strting either from X, or from Y, nd pling O (the letter for the vertex) in the middle. Rememer to ple the ngle symol ( ) efore the letters. Nme the ngle y strting either from, or from C, nd pling (the letter representing the vertex) in the middle. Rememer to ple the ngle symol ( ) efore the letters. WorKeD exmple 8 Drw the ute ngle DG. think 1 Construt ny ute ngle (unless speified otherwise). 2 Letter D is in the middle, so ple it t the vertex. 3 Ple letter t one rm of the ngle nd letter G t the other. (Note tht it does not mtter whih rm is represented y D nd GD.) C WrIte XOY or YOX C or C Exerise 5.4 Types of ngles nd nming ngles InDIVIDuL PtHWys PrCtIse Questions: 1 11, 13, 15 ConsoLIDte Questions: 1 13, 15 Individul pthwy intertivity int-4333 DrW D mster Questions: 1 15 G 158 Mths Quest 7

16 mesurement ND geometry FLUENCY 1 WE6 Clssify eh of the following ngles ording to its size d 156 e 180 f 62 g h 90 i 4.8 j 360 k 100 l 45 m 82 n 270 o p The following list gives vlues of prtiulr ngles. 3, 45, 65, 123, 69, 234, 90, 360, 300, 270, 165, 210, 180 How mny of these re ute ngles? List them. How mny of these re otuse ngles? List them. Is there right ngle in the list? d Is there stright ngle? e How mny reflex ngles re there? List them. f Is there full revolution? Wht is its vlue? g Wht is the omplement of 65? h Wht is the supplement of 123? 3 Consider the following digrms nd write down the type of ngle shown in eh se. d e f 4 WE7 Nme eh of the following ngles. X P V D S U X d G Q P R Z e P F H f O Z R Topi 5 Geometry 159

17 mesurement nd geometry 5 WE8 Drw eh of the following ngles s speified in rkets. CDE (ute) TRE (otuse) QS (stright) d FGH (reflex) e KJF (right) f NV (ute) 6 Nme eh ute nd otuse ngle in the following digrms. X d g R P D F O O N Y M T e U O G K V S R M understnding 7 Stte the type of ngle shown in eh photogrph. h P Y J H X Q O P Q O W N L O f T R O d K P V W O Z P K P 160 Mths Quest 7

18 mesurement nd geometry 8 Stte the type of ngle tht the minute hnd of lok sweeps while moving for the period of: 10 minutes 15 minutes 20 minutes d 30 minutes e 45 minutes f 1 hour. 9 t vrious times of the dy the hnds of lok form two ngles: the inside ngle (the smller one) nd the outside ngle (the lrger one). Stte the type of the inside ngle formed y the hnds of the lok t: i 1 o lok ii 20 minutes to 12 iii 6 o lok iv 9 o lok v qurter to 4 vi hlf pst 10. Write two different times when the inside ngle of the lok is: i ute ii otuse iii right. 10 ngles re very importnt in engineering nd rhiteture. Three ngles hve een mrked in Figure 1. Clssify eh ngle. Use protrtor to mesure eh ngle. With prtner, lote s mny other ngles s you n tht re the sme size s the mrked ngles 1, 2 nd In Figures 2 nd 3, ute, right ngled nd otuse ngles hve een used. i Nme n ute ngle. ii Nme two right ngles. iii Nme n otuse ngle. iv Cn you nme reflex ngle? P C M N O C Q D R G F I H D H U T E F Figure 2 2 Figure 1 1 E W X Y Figure 3 Topi 5 Geometry 161

19 mesurement nd geometry resoning 12 Give resons for the following sttements. The sum of two ute ngles n e n ute ngle. The sum of two ute ngles n e right ngle. The sum of two ute ngles n e n otuse ngle. d The sum of two ute ngles n e reflex ngle. 13 Cn you onstrut full revolution using one 90 ngle, one stright ngle, nd two ute ngles? Construt digrm to represent the full revolution in prt. ProLem solving 14 Wthmkers tke prtiulr re tht the ngles etween the hnds of n nlogue lok represent the time urtely. Wht is the smller ngle etween the hour nd minute hnds when n nlogue lok displys the time s ten pst nine? 15 Lel eh ngle mrked in the following digrm with the word desriing the type of ngle it is. CH HLLenge 5.1 Digitl do WorkSHEET 5.1 do Mths Quest 7

20 MESUREMENT ND GEOMETRY 5.5 Tringles Types of tringles Every tringle hs three ngles nd three sides. Cpitl letters of the English lphet re used t eh vertex (in ny order) to nme tringles. The symol Δ is used in ple of the word tringle. This tringle n e nmed ΔC. (It n lso e nmed ΔC, ΔC, ΔC, ΔC or ΔC.) Identil mrks on the sides of the tringle re used to indite tht the sides hve the sme length. Likewise, identil mrks on the ngles indite tht the ngles re equl in size. Clssifying tringles ording to the length of their sides ording to their side lengths, tringles n e lssified s equilterl, isoseles or slene. n equilterl tringle hs ll sides equl in length. (ll ngles of n equilterl tringle re lso equl in size.) n isoseles tringle hs two sides of equl length. (The ngles djent to the unequl third side re lso equl in size.) slene tringle hs no equl sides. (ll ngles in slene tringle re different in size.) WORKED EXMPLE 9 Clssify eh of these tringles ording to the lengths of its sides. THINK C M N P Sides nd C hve identil mrkings on them, whih indite tht they re of equl length. So ΔC hs 2 equl sides. Clssify it ordingly. The 3 sides of ΔMNP hve identil mrkings on them, whih mens tht ll 3 sides re equl in length. Clssify this tringle. P R WRITE S ΔC is n isoseles tringle. ΔMNP is n equilterl tringle. C Digitl dos SkillSHEET Clssifying tringles ording to the length of their sides do-6454 SkillSHEET Clssifying tringles ording to the size of their ngles do-6455 Topi 5 Geometry 163

21 mesurement nd geometry ll 3 sides of ΔPRS re mrked differently. Therefore, no sides in this tringle re equl in length. Use this informtion to lssify the tringle. ΔPRS is slene tringle. Clssifying tringles ording to the size of their ngles ording to the size of their ngles, tringles n e lssified s eing right-ngled, ute-ngled, or otuse-ngled. right-ngled tringle hs one of its ngles equl to 90 (one of its ngles is right ngle). smll squre in the orner mrks the right ngle. WorKeD exmple 10 ngles in tringle n ute-ngled tringle hs ll ngles smller thn 90 (ll three ngles re ute). The sum of the three ngles in ny tringle is lwys mkes stright ngle n otuse-ngled tringle hs one ngle greter thn 90 (one ngle is otuse). Clssify eh of the tringles in Worked exmple 9 ording to the size of its ngles. think In ΔC, C is mrked s the right ngle, so lssify it ordingly. In ΔMNP ll ngles re less thn 90, so lssify this tringle. In ΔPRS, PRS is greter thn 90 ; tht is, it is otuse. Use this informtion to lssify the tringle. WrIte ΔC is right-ngled tringle. ΔMNP is n ute-ngled tringle. ΔPRS is n otuse-ngled tringle. + + = Mths Quest 7

22 mesurement nd geometry WorKeD exmple 11 Find the vlue of the pronumerl in eh of the following tringles think 58 1 The sum of the three ngles (, 35 nd 58 ) must e 180. Write this s n eqution. d WrIte = Simplify y dding 35 nd 58 together = Use inspetion or ktrking to solve for The sum of the three ngles (d, 37 nd 42 ) must e 180. Write this s n eqution. = = 87 d = Simplify y dding 37 nd 42 together. d + 79 = Use inspetion or ktrking to solve for d. d 101 WorKeD exmple d Find the vlue of the pronumerl in the following tringle. think 74 C h 1 The mrkings on the digrm indite tht ΔC is isoseles with = C. Therefore, the ngles t the se re equl in size; tht is, C = C = 74. WrIte C = C = 74 d = d = ll 3 ngles in tringle must dd up to 180. C + C + C = 180 h = 180 Topi 5 Geometry 165

23 mesurement nd geometry 3 Simplify. h = Solve for h, tht is, sutrt 148 from 180. h = h = 32 WorKeD exmple 13 Find the vlue of the pronumerl in the following tringle. M think Interior nd exterior ngles of tringle The ngles inside tringle re lled interior ngles. If ny side of tringle is extended outwrds, the ngle formed is lled n exterior ngle. The exterior ngle nd the interior ngle djent (next) to it dd up to 180. The sum of the two interior opposite ngles is equl to the exterior ngle. Interior ngles N 40 P Exterior ngle WrIte 1 From the digrm we n see tht ΔMNP is NMP = NPM = isoseles with MN = NP. Hene, NPM = NMP =. 2 Form n eqution y putting the sum of the ngles on one side nd 180 on the other side of the equls sign. NMP + NPM + MNP = = Simplify y olleting like terms = Use inspetion or ktrking to solve for. 2 = = = = Interior ngle Exterior ngle C C + CD = 180 nd C + C = CD. D 166 Mths Quest 7

24 mesurement nd geometry WorKeD exmple 14 Find the vlue of the pronumerls in the following digrm. n D think 125 Exerise 5.5 Tringles InDIVIDuL PtHWys PrCtIse Questions: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16 p 83 1 C (ngle p) together with its djent exterior ngle ( D) dd up to 180. Furthermore, D = 125. So, form n eqution. 2 Solve for p, tht is sutrt 125 from The interior ngles of ΔC dd up to 180. Identify the vlues of the ngles nd form n eqution. C WrIte ConsoLIDte Questions: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16,, 18, 20, 23, 24 Individul pthwy intertivity int-4334 C = p; D = 125 C + D = 180 p = 180 p = p = 55 C + C + C = 180 C = 83 ; C = p = 55 ; C = n n = Simplify y dding 83 nd 55. n = Solve for n; tht is, sutrt 42 from 180. n = n = 42 mster Questions: 1, 2, 3, 4, 5, 6, e, f, 7, 8, 9, f, FLuenCy 1 WE9 Nme eh of the following tringles using the pitl letters, then lssify eh tringle ording to the lengths of its sides. L reflection How n you prove the sttement: The sum of the opposite interior ngles is equl to the exterior ngle? C M N Topi 5 Geometry 167

25 mesurement ND geometry X Y d P Z Q R e R f E S T 2 WE10 Clssify eh of the tringles in question 1 ording to the size of its ngles. 3 dd side nd ngle mrkings to these digrms to show tht: ΔRST is n equilterl tringle ΔUVW is n isoseles tringle R S T ΔPQR is slene tringle d ΔMNP is right-ngled tringle P Q R e ΔC is right-ngled nd isoseles tringle C F U M V W f ΔMNO is right-ngled nd slene tringle. 4 MC Whih of these tringles is n equilterl tringle? C M N P N O G D E 168 Mths Quest 7

26 mesurement ND geometry Whih of these tringles is not slene tringle? C D 5 MC Whih of these tringles is oth right-ngled nd slene? C D Whih of these tringles is oth ute-ngled nd isoseles? C D E E E 6 WE11 Find the vlue of the pronumerl in eh of the following tringles x 30 g t d 60 e 33 f k 60 f z 60 Topi 5 Geometry 169

27 mesurement ND geometry 7 Find the vlue of the pronumerl in eh of the following right-ngled tringles d 40 8 WE12 Find the vlue of the pronumerl in eh of the following tringles, giving resons WE13 Find the vlue of the pronumerl in eh of the following tringles, giving resons. d t u e 10 WE14 Find the vlue of the pronumerls in the following digrms. 130 p 50 n k e 57 f f d 125 s d 28 n 70 p x y 130 t e f n 170 Mths Quest 7

28 mesurement nd geometry g m h t m 120 t 56 understnding 11 Wht types of tringles n you see in the sndwihes on the right? 12 Write down three ute tringles you n see round you. 13 Find one exmple in your lssroom or home of eh of the six types of tringles desried in this hpter. Desrie lerly where the tringle ours, drw the tringle nd lssify it ording to oth side nd ngle types. 14 In the following tringle: how mny equilterl tringles n you find? how mny right-ngled tringles n you find? how mny isoseles tringles n you find? 15 Use your ruler, penil nd protrtor to urtely drw: n equilterl tringle with side lengths 6 m nd ll ngles 60 n isoseles tringle with two sides whih re 6 m eh with 40 ngle etween them right-ngled tringle whose two short sides re 6 m nd 8 m. (How long is the longest side?) d slene tringle with two of the sides mesuring 4 m nd 5 m nd n ngle of 70 etween the two sides. 16 Find the missing ngle in eh of the tringles mrked on the following photogrphs. 70 d p 60 p k Topi 5 Geometry 171

29 mesurement ND geometry 17 n isoseles tringle hs 2 ngles of 55 eh. Find the size of the third ngle. n isoseles tringle hs 2 ngles of 12 eh. Find the size of the third ngle. Two ngles of tringle re 55 nd 75 respetively. Find the third ngle. d Two ngles of tringle re 48 nd 68 respetively. Find the third ngle. 18 Use ruler nd protrtor to onstrut eh of the following tringles. i n isoseles tringle with se of 4 m nd equl ngles of 50 eh. ii n isoseles tringle with two sides whih re 5 m eh nd two equl ngles whih re 45 eh. On your digrms lel the size of eh ngle. Clssify the tringles ording to the size of their ngles. 19 For eh of the following sets of 3 ngles, stte whether or not it is possile to onstrut tringle with these ngles. Give reson for your nswer. 40, 40, , 60, 70 45, 55, 85 d 111, 34.5, Explin in your own words why it is impossile to onstrut tringle with 2 otuse ngles. RESONING 21 Cut eighteen strips of pper three eh of the following six lengths: 3 m, 4 m, 5 m, 6 m, 8 m nd 10 m. Use these strips to help you nswer the following questions. Using omintion of three of these strips (without ending them), how mny different tringles n you mke? Wht type of tringle is eh one? How mny re there of eh kind? re there ny similr tringles (sme shpe, ut different size)? d Mesure the size of the ngles. Wht generl reltionship n you oserve etween the lengths of the sides of tringle nd the size of the ngles? e Formulte generl sttement whih would enle you to determine whether it is possile to form tringle from three given side lengths. Why is it not possile in some ses? 22 Construt the isoseles tringle C with = C = 80. Extend the line C to the right nd lel the other end D where D = nd form the ΔD. Clulte the mgnitude of D. d Extend the line CD to the right nd lel the other end E where DE = D nd form the ΔDE. e Clulte the mgnitude of E. f Repet this proess gin. g Wht is the mgnitude of the lst ngle? h Is there pttern? Explin your nswer. PROLEM solving 23 ΔMNP is n isoseles tringle with MN = MP nd MNP = 72. Clulte the mgnitudes of MPN nd NMP. ΔMNP is n isoseles tringle with MN = MP nd NMP = 72. Clulte the mgnitudes of MPN nd MNP 172 Mths Quest 7

30 mesurement nd geometry 24 ΔC hs its verties on Ferris wheel s shown in the digrm. C is dimeter of the wheel euse it psses through the entre of the wheel. Use protrtor to mesure the three ngles of the tringle nd reord these vlues in tle like the one shown. Tringle C Type of tringle 1 2 Construt nother tringle y moving point on the irumferene of the irle. Keep C unhnged. Mesure the new ngles nd reord these vlues in your tle. d Construt nother tringle y moving points nd C on the irumferene of the irle to form nother dimeter. Keep point unhnged. e Mesure the new ngles nd reord these vlues in your tle. f Construt nother tringle y moving the three points, nd C on the irumferene of the irle. The only ondition is to ensure tht C is dimeter of the irle. g Wht onlusion n you drw from this tivity? CHLLenge 5.2 C Topi 5 Geometry 173

31 mesurement ND geometry 5.6 Qudrilterls nd their properties Types of qudrilterls qudrilterl is 2-dimensionl losed shpe with four stright sides. Qudrilterls n e divided into two mjor groups: prllelogrms nd other qudrilterls. Prllelogrms re qudrilterls with oth pirs of opposite sides eing prllel to eh other. ll retngles, squres nd rhomi (dimonds) re prllelogrms. Other qudrilterls (non-prllelogrms) inlude trpeziums, kites nd irregulr qudrilterls. Prllelogrms Shpe Properties Prllelogrm Retngle Rhomus Squre Opposite sides re equl in length. Opposite ngles re equl in size. Opposite sides re equl in length. ll ngles re the sme nd equl 90. ll sides re equl in length. Opposite ngles re equl in size. ll sides re equl in length. ll ngles re the sme nd equl 90. Other qudrilterls Shpe Properties Trpezium Kite One pir of opposite sides is prllel. Two pirs of djent (next to eh other) sides re equl in length. One pir of opposite ngles (the ones tht re etween the sides of unequl length) re equl. Irregulr qudrilterl This shpe does not hve ny speil properties. 174 Mths Quest 7

32 mesurement nd geometry WorKeD exmple 15 Nme the following qudrilterls, giving resons for your nswers. think The mrkings on this qudrilterl indite tht ll sides re equl in length nd ll ngles equl 90. Clssify the qudrilterl y finding the mthing desription in the tle. The rrows on the sides of this qudrilterl indite tht there re two pirs of prllel sides nd hene it is prllelogrm. Chek the desriptions in the tle to see if it is prtiulr type of prllelogrm. ngles in qudrilterl The sum of ngles in ny qudrilterl is 360. WorKeD exmple 16 Qudrilterls re mde up of 2 tringles Sum of ngles in 1 tringle = tringles = = d = 360 d Find the vlue of the pronumerl in the digrm. think WrIte The given qudrilterl is squre, sine ll sides re equl nd ll ngles re 90. The given qudrilterl is prllelogrm, sine it hs two pirs of prllel sides. 1 The sum of the ngles in qudrilterl is 360. So, express this s n eqution. WrIte = Simplify y dding 120, 80 nd = Solve to find the vlue of ; tht is, sutrt 275 from 360. = = 85 Topi 5 Geometry 175

33 mesurement nd geometry WorKeD exmple 17 Find the vlue of the pronumerl in the digrm, giving reson for your nswer. x 72 think WrIte ording to the mrkings, the opposite sides of the given qudrilterl re prllel nd equl in length. Therefore, this qudrilterl is prllelogrm. In prllelogrm opposite ngles re equl. So, stte the vlue of the pronumerl. WorKeD exmple 18 Find the vlue of the pronumerls in the digrm t right. think 1 Form n eqution y writing the sum of the ngles on one side nd 360 on the other side of n equls sign. 2 The qudrilterl shown in the digrm is kite. ngle t nd ngle 136 re the ngles etween unequl sides nd therefore must e equl in size. Opposite ngles in prllelogrm re equl in size. Therefore, x = 7 2. WrIte k + t = 360 t = 136 (The figure is kite.) 3 Reple t in the eqution with 136. k = Simplify. k = 360 t k 5 Solve to find the vlue of k, tht is, sutrt 322 from 360. k = k = Mths Quest 7

34 mesurement nd geometry Exerise 5.6 Qudrilterls nd their properties InDIVIDuL PtHWys PrCtIse Questions: 1, 2, 3, 4, 5, 6, 7, 9, 12, 14, 18, 21 ConsoLIDte Questions: 1, 2, 3,, d, g, h, 4, d, f, 5, 6, 7, 9, 10, 12, 14, 16, 17, 18, 20, 21 Individul pthwy intertivity int-4335 mster Questions: 1, 2, 3, d, e, h, 4, d, 5, 6, 7, 8 22 FLuenCy 1 WE15 Nme the following qudrilterls, giving resons for your nswers. d e f 2 MC This qudrilterl is : squre retngle C kite D prllelogrm e rhomus This qudrilterl is : trpezium prllelogrm C rhomus D kite e squre This qudrilterl is : trpezium squre C kite D rhomus e prllelogrm reflection Wht would e n esy wy to rememer the sum of ngles in qudrilterls? Topi 5 Geometry 177

35 mesurement ND geometry 3 WE16 Find the vlue of the pronumerl in eh of the following digrms. t d 18 e g i p 127 t k 93 s f h 20 m t WE17 Find the vlue of the pronumerl in eh of the following digrms, giving resons for your nswers. m t 78 u d 108 e 63 f 73 f p z 178 Mths Quest 7

36 mesurement ND geometry 5 WE18 Find the vlue of the pronumerls in eh of the following digrms. 98 d m m t d m p e 75 6 MC The vlue of t in the digrm t right is: C 222 D 138 E MC The vlue of r in the digrm t right is: C 234 D 126 E 63 UNDERSTNDING 8 Stte whether eh of the following sttements is true or flse. ll squres re retngles. ll squres re rhomuses. ll retngles re squres. d ny rhomus with t lest one right ngle is squre. e retngle is prllelogrm with t lest one ngle equl to MC retngle is qudrilterl euse: it hs 4 right ngles it hs 2 pirs of prllel sides C its opposite sides re equl in length D it hs 4 stright sides E it hs 2 pirs of prllel sides nd 4 right ngles 10 Drw 4 equilterl tringles with side lengths 4 m nd ut them out. Use 2 of these tringles to mke rhomus. Drw your solution. Use 3 of these tringles to mke trpezium. Drw your solution. Use ll 4 tringles to mke prllelogrm. Drw your solution. 11 Copy nd ut out the set of shpes shown t right. rrnge the shpes to form squre. Drw your solution. x t f 64 m r 115 t 117 t p 42 n Topi 5 Geometry 179

37 mesurement ND geometry 12 In your house, find n exmple of eh type of qudrilterl disussed in this setion. Write down the type of qudrilterl nd where you found it. 13 This photogrph shows the roof of gzeo. 119 Clulte the vlue of p. p 14 Find the size of the otuse ngle in the following kite. 15 Two ngles in prllelogrm re 45 nd 135. Find the other 2 ngles. 16 Tom mesures 2 ngles of kite t 60 nd 110, ut forgets whih ngle is whih. Drw 3 different kites tht Tom my hve mesured, showing the size of ll ngles in eh digrm. 17 For eh of the following sets of 4 ngles, deide whether it is possile to onstrut qudrilterl. Explin your nswer. 25, 95, 140, , 80, 99, Three ngles of qudrilterl re 60, 70 nd 100. Wht is the size of the fourth ngle of this qudrilterl? How mny qudrilterls with this set of 65 ngles re possile? Construt one qudrilterl with the given ngle sizes in your ook. (The hoie of the length of the sides is yours.) RESONING 19 Vl nd Peter wnt to reple their front gte with nother of the sme design. To hve this gte mde, they need to supply digrm of it with ll mesurements nd ngles shown. Study the photogrph of Vl nd Peter s gte nd use it to help you nswer the following questions. 8 m m 60 m 1 m 180 Mths Quest 7

38 mesurement nd geometry There re 4 different shpes formed y the metl rs of the gte. How mny different types of tringles re there? Cn you nme them? How mny types of qudrilterls re there? Nme them. Drw digrm of the gte showing the length mesurements nd the one ngle tht is given. d Use this ngle to lulte ll the remining ngles in the digrm. e Using ruler nd protrtor, design fene tht is to e onstruted using metl rs. Inlude different tringles nd qudrilterls to mke your design s interesting s possile. Write short report desriing the shpes nd importnt ngles you hve used in your design to ssist in the onstrution of the fene. 20 Consider the digrm shown. Prove tht the sum of ll ngles in qudrilterl is 360. D s m r ProLem solving 21 Wht is the ngle sum (sum of ll the ngles) of degon? 22 Drw qudrilterl, PQRS. Lel ngle P s 100 nd ngle R s right ngle. ngle Q is two-thirds of ngle S. Determine the mesurement of ngle S. 5.7 Prllel nd perpendiulr lines n Vertilly opposite nd djent ngles When two stright lines interset, they form four ngles with ommon vertex. The ngles tht re opposite eh other re lled vertilly opposite ngles. Vertilly opposite ngles re equl in size. The ngles next to eh other re lled djent ngles. (djent ngles shre n rm.) In the following digrm there re two pirs of vertilly opposite ngles. O = DOC nd OC = OD There re 4 pirs of djent ngles. O is djent to OC. OC is djent to COD. COD is djent to OD. OD is djent to O. Prllel lines Prllel lines re indited in digrm y lines with rrows on them. line interseting pir (or set) of prllel lines is lled trnsversl. Cutting prllel lines y trnsversl retes series of ngles tht re relted in numer of wys. The symol used to men is prllel to is. q C p D O C Prllel lines Intertivities Vertilly opposite ngles int-0814 Corresponding ngles int-0815 lternte ngles int-0816 Co-interior ngles int-0817 Trnsversl Topi 5 Geometry 181

39 mesurement nd geometry Corresponding ngles The following two digrms show two ngle, nd, positioned elow the prllel lines to the right of trnsversl. When oth ngles re on the sme side of the trnsversl (oth to the left, or oth to the right of it) nd re either oth ove, or oth elow the prllel lines, suh ngles re lled orresponding ngles. Corresponding ngles re equl in size. Thus, in the digrm t right =. The position of orresponding ngles is esy to rememer y ssoiting it with n F shpe. Co-interior ngles (or llied ngles) The digrm t right shows two ngles, nd, positioned etween the prllel lines, on the sme side (to the right) of the trnsversl. Suh ngles re lled o interior ngles or llied ngles. Co-interior ngles re supplementry; tht is, they dd up to 180. Thus, in the digrm t right + = 180. The position of the o-interior ngles is esy to rememer y ssoiting it with C shpe. lternte ngles The digrm t right shows two ngles, nd, positioned etween the prllel lines nd on lternte sides of the trnsversl. Suh ngles re lled lternte ngles. lternte ngles re equl in size. Thus, in the digrm t right, =. The position of lternte ngles is esy to rememer y ssoiting it with Z shpe. Clulting ngles ssoited with prllel lines ngle reltionships ssoited with prllel lines n e used to find the size of missing ngles, s shown in the following worked exmples. It is importnt to rememer tht Z, F or C shpes my e upside down or kwrds in some digrms. WorKeD exmple 19 For the digrm t right: stte the type of ngle reltionship find the vlue of the pronumerl. m Mths Quest 7

40 mesurement nd geometry think 1 Study the digrm: whih shpe Z, F or C would inlude oth ngles tht re shown? Copy the digrm into your workook nd highlight the pproprite shpe. WrIte m 45 2 Stte the nme of the ngles suggested y C shpe. 1 Co-interior ngles dd to 180. Write this s n eqution. Shown ngles re o-interior. m + 45 = Solve for m. m = = 135 WorKeD exmple 20 Find the vlue of the pronumerl in the digrm shown, giving resons. think 1 The two ngles shown re neither C, Z nor F ngles. So we must find some other ngle first, tht will enle us to find the size of ngle x. (This other ngle must e in ertin reltion with oth given ngles.) Drw n F shpe tht inludes 120 ngle. The other ngle in n F shpe is relted to oth given ngles: it is orresponding to the 120 ngle nd it is supplementry to ngle x. Cll this ngle y. 2 Stte the size of ngle y, speifying the reson (the ngle reltion). 3 The ngles x nd y re supplementry (dd to 180 ). Stte this s n eqution. 4 Sustitute the vlue of y into the eqution. WrIte 120 y x y = 120 (s orresponding ngles re equl) x + y = 180 (s ngle sum of stright line is 180 ) x = Solve for x. x = = 60 x 120 Topi 5 Geometry 183

41 mesurement nd geometry Testing for prllel lines If 2 lines nd trnsversl interset to form ny of the following ngles: orresponding ngles lternte ngles o-interior ngles then the two lines re prllel. WorKeD exmple 21 re the following lines nd prllel? Give resons for your nswer. think 1 The given ngles re equl to eh other. They re oth Stte the nme of the ngles suggested y Z shpe. These ngles re lternte. 3 Sine the 2 ngles re lternte nd equl, then the lines nd must e prllel. Perpendiulr lines Perpendiulr mens t right ngles. When two line segments meet t right ngles they re sid to e perpendiulr. Tht is, the ngle etween them is 90. In the following digrm, the ngles OX nd XO re eh right ngles, whih mens tht the line segments nd OX re perpendiulr. To write the line segments nd OX re perpendiulr using mthemtil symols, simply write OX. The symol mens is perpendiulr to. WrIte O X 184 Mths Quest 7

42 mesurement nd geometry Exerise 5.7 Prllel nd perpendiulr lines InDIVIDuL PtHWys PrCtIse Questions: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16 FLuenCy 1 Copy nd omplete this tle. ConsoLIDte Questions: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16 Individul pthwy intertivity int-4336 mster Questions: 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16 Digrm Type of ngle reltion ssoited shpe Rule Corresponding lternte 2 Copy the following digrm into your workook. Clerly drw n F shpe on your digrm nd lel the ngle orresponding to the one tht is mrked. Copy the following digrm into your workook. Clerly drw Z shpe on your digrm nd lel the ngle lternte to the mrked ngle. Copy the following digrm nd lel the ngle o-interior to the mrked ngle. Clerly drw C shpe on your digrm. 3 Mth eh digrm with the pproprite nme from the four options listed. Digrm C D Z re equl in size. dd up to 180. Nme Co-interior ngles (C) Corresponding ngles (F) lternte ngles (Z) None of the ove d reflection Cn you think of wy to rememer the different types of ngles ssoited with prllel lines nd their speil properties? Digitl do Investigtion Investigte prllel nd perpendiulr lines do-6456 Topi 5 Geometry 185

43 mesurement ND geometry 4 In the following digrm, list ll pirs of: vertilly opposite ngles orresponding ngles o-interior ngles d lternte ngles. 5 MC In the following digrm: whih ngle is vertilly opposite to ngle p? k m C r D q E t whih ngle is orresponding to ngle p? k m C r D q E t whih ngle is o-interior to ngle ṗ? k m C r D q E n d whih ngle is lternte to ngle p? k m C r D q E n e nme two ngles djent to ngle k. 6 WE19 For eh of the following digrms: i stte the type of ngle reltionship ii find the vlue of the pronumerl. d p t e 45 q m 7 WE20 Find the vlue of the pronumerls in eh of the following digrms, giving resons. d 116 y g 68 e 38 z h 62 f f n s d e f g h r n t p k m j q 65 k Mths Quest 7

44 mesurement nd geometry 8 WE20 Find the vlue of the pronumerls in the following digrms. i ii iii iv H G F E F 123 G H 80 J z z D C D E C v D E U T x X 160 F 137 Nme ll the perpendiulr lines in the preeding digrms in prt. understnding 9 If the ngle llied to x is 135, find the size of ngle x. 10 If the ngle orresponding to y is 55, find the size of ngle y. 11 hill is t n ngle of 30 to the horizontl. fene is put in, onsisting of riling prllel to the ground nd vertil fene posts. Find the ngle, p, etween the top of the fene post nd the ril. 12 Two gtes onsist of vertil posts, horizontl struts nd digonl ems. Find the ngle,, s shown in the following gtes. 40 V P p W resoning 13 WE21 Is the line prllel to the line CD? Explin your nswer C H G Q R S vi F V G T E 62 M S 50 q q W 30 H N R p y Y Z P D C Q C 133 D Topi 5 Geometry 187

45 mesurement nd geometry 14 Perspetive drwing lso llows us to rete the illusion of looking t the sme ojet from numer of different ngles. For exmple, onsider the following simple ox. Figures nd re views of the ue from different ngles. In ll ses the top of the ox hs een shded. Figure Figure Find the vnishing point in eh digrm, y extending the edges in the digrm until they meet. True or Flse? Explin your nswers. i The viewer is ove the ox in Figure. ii The viewer is ehind the ox in Figure. iii The viewer of Figure is proly tller thn the viewer of Figure. ProLem solving 15 In the digrm shown lulte the mgnitude of the ute ngle PQR nd the reflex ngle PQR. P R Look t the following optil illusions. re ny of the lines prllel? Explin your nswer. Q Digitl do WorkSHEET 5.2 do Mths Quest 7

46 mesurement nd geometry ONLINE ONLY 5.8 Review The Mths Quest Review is ville in ustomisle formt for students to demonstrte their knowledge of this topi. The Review ontins: Flueny questions llowing students to demonstrte the skills they hve developed to effiiently nswer questions using the most pproprite methods Prolem solving questions llowing students to demonstrte their ility to mke smrt hoies, to model nd investigte prolems, nd to ommunite solutions effetively. summry of the key points overed nd onept mp summry of this hpter re ville s digitl douments. int-2591 int-2592 int-3166 Lnguge ute ngle ute-ngled tringle djent ngles lternte ngles ngle o-interior ngles (llied) omplementry ngles equilterl tringle exterior ngle interior ngle irregulr polygon isoseles tringle Link to ssesson for questions to test your rediness For lerning, your progress s you lern nd your levels of hievement. ssesson provides sets of questions for every topi in your ourse, s well s giving instnt feedk nd worked solutions to help improve your mthemtil skills. kite line segment otuse ngle otuse-ngled tringle prllel prllelogrm perpendiulr perigon protrtor qudrilterl retngle refl ex ngle Review questions Downlod the Review questions doument from the links found in your eookplus. revolution rhomus right ngle right-ngled tringle slene squre stright ngle supplementry ngles trnsversl trpezium vertex vertilly opposite ngles Link to SpyClss, n exiting online gme omining omi ook style story with prolem-sed lerning in n immersive environment. Join Jesse, Toy nd Dn nd help them to tkle some of the world s most dngerous riminls y using the knowledge you ve gined through your study of mthemtis. Topi 5 Geometry 189

47 <InVestIgtIon> For rich tsk or <mesurement nd geometry> For PuZZLe InVestIgtIon rich tsk S M PL E EV L U T IO N O N LY Tngrms 190 Mths Quest 7 05Geometry.indd /05/16 3:06 PM

48 mesurement nd geometry Instrutions Tre the following tngrm piees onto rdord nd ut out eh piee urtely. 1 Using only the two smll tringles, mke nd drw the following shpes, showing the joins in your ook. Squre Tringle Prllelogrm 2 Using the squre, the two smll tringles, the prllelogrm nd the medium tringle, mke nd drw the following shpes. Squre Hexgon Trpezium d Prllelogrm 3 Using ll the piees, mke nd drw the following shpes. Tringle Retngle Hexgon 4 Use ll of the piees to mke the following shpes nd the other shpes shown ove. Roket The letter E Topi 5 Geometry 191

49 <InVestIgtIon> mesurement nd For geometry rich tsk or <mesurement nd geometry> For PuZZLe CoDe PuZZLe Money of the world efore the euro Complete the ngles round this pge on the selines given y drwing in the extr lines to form the ngles stted. The numers on the selines nd the irled letters tht your lines pss through give the puzzle s nswer ode F D I U Y J L P N R O E S C K W G T Mths Quest 7

50 mesurement nd geometry tivities 5.2 mesuring ngles Digitl dos SkillSHEET (do-6447) Reding the sle on protrtor SkillSHEET (do-6448) Compring the size of ngles SkillSHEET (do-6449) Estimting the size of n ngle SkillSHEET (do-6450) Mesuring ngles with protrtor Intertivities Digitl protrtor (int-2348) IP intertivity 5.2 (int-4331) Mesuring ngles 5.3 Construting ngles with protrtor Digitl do SkillSHEET (do-6451) Construting ngles with protrtor Intertivity IP intertivity 5.3 (int-4332) Construting ngles with protrtor 5.4 types of ngles nd nming ngles Digitl dos SkillSHEET (do-6452) Clssifying ngles SkillSHEET (do-6453) Nming ngles WorkSHEET 5.1 (do-1755) Intertivity IP intertivity 5.4 (int-4333) Types of ngles nd nming ngles 5.5 tringles Digitl dos SkillSHEET (do-6454) Clssifying tringles ording to the length of their sides to ess eookplus tivities, log on to SkillSHEET (do-6455) Clssifying tringles ording to the size of their ngles Intertivity IP intertivity 5.5 (int-4334) Tringles 5.6 Qudrilterls nd their properties Intertivity IP intertivity 5.6 (int-4335) Qudrilterls nd their properties 5.7 Prllel nd perpendiulr lines Intertivities Vertilly opposite ngles (int-0814) Corresponding ngles (int-0815) lternte ngles (int-0816) Co-interior ngles (int-0817) IP intertivity 5.7 (int-4336) Prllel nd perpendiulr lines Digitl dos Investigtion (do-6456) Prllel nd perpendiulr lines WorkSHEET 5.2 (do-6618) 5.8 review Intertivities Word serh (int-2591) Crossword (int-2592) Sudoku (int-3166) Digitl dos Topi summry (do-10726) Conept mp (do-10727) Topi 5 Geometry 193

51 mesurement ND geometry nswers TOPIC 5 Geometry 5.2 Mesuring ngles d 13 e 142 f 107 g 162 h 103 i 240 j 201 k 316 l Disuss with your teher s nswers will vry d 180 e 77 f 63 5 Diffiult, s ngle is pproximtely 7 Diffiult 6,,, e, f, h vrious nswers d, g Digrm Mesured ngle (reflex ngle) nd 150 (otuse ngle), 180 (stright ngle) nd 180 (stright ngle), 150 (otuse ngle) nd 210 (reflex ngle), 15 (ute ngle) nd 345 (reflex ngle), 108 (otuse) nd 252 (reflex ngle), 114 (otuse ngle) nd 246 (reflex ngle) minutes = 6 ; hours = 30 8 s the kite moves from point to point, the ngle inreses from 0 to 90. s the kite moves from point to point C, the ngle dereses from 90 to Construting ngles with protrtor d g e h f e 140 f g 127 h d 260 e 243 f 192 g h C d 349 e 328 f 300 g h d Mths Quest 7

52 mesurement ND geometry d 187 e , 12 nd 90 8 Construt the orresponding ute ngle nd then lel the reflex ngle. 9 Vrious nswers possile. Vrious nswers possile. 10 ll tringles should e identil. If two equl ngles re drwn t the ends of stright line of set length, there is only one size tringle tht forms. 11 Drw ny tringle on n 4 sheet of pper. Vrious mgnitudes possile. d 180 e The sum of the ngles of tringle is Types of ngles nd nming ngles 1 ute Reflex Otuse d Otuse e Stright f ute g Reflex h Right i ute j Revolution k Otuse l ute m ute n Reflex o ute p Reflex 2 4; 3, 45, 65, 69 2; 123, 165 Yes, 90 d Yes, 180 e 4; 234, 300, 270, 210 f Yes, 360 g 25 h 57 3 Right Stright line Otuse d ute e Reflex f Revolution 4 XUV or VUX PQR or RQP SDX or XDS d GPZ or ZPG e PFH or HFP f ZRO or ORZ 5 nswers will vry; however, in eh se, the middle letter should e t the vertex of the ngle. D Q C E S d T H G R F E e K f J F 6 ute: NOM, MOY; otuse: MOX ute: PQS, RQO; otuse: SQR, SQO, RQP ute: MLN, NLO, OLP, PLK, NLP, OLK; otuse: MLO, MLP, NLK d ute: POR, ROT, TOU, UOV; otuse: POU, ROU, ROV e ute: YOX, WOQ, POQ; otuse: XOW, YOP, YOW, XOP, YOQ, POW f ute: VOT, TOR, ROP, POZ, ZOW, WOV; otuse: VOR, TOP, ROZ, POW, ZOV, WOT g ute: FOG, GOK; otuse: DOG h ute: JOH, KOH, POK; otuse: JOK, HOP 7 Right ute Revolution d Otuse 8 ute Right Otuse d Stright e Reflex f Revolution 9 i ute ii Otuse iii Stright iv Right v Otuse vi Otuse Vrious nswers 10 ngle 1: otuse; ngle 2: ute; ngle 3: stright ngle 1: 140 ; ngle 2: 48 ; ngle 3: 180 Vrious nswers 11 In eh se the middle letter should e t the vertex of the ngle. i ute: HUT ii Right: C, GHI iii Otuse: DEF iv Reflex: MNO i DEF ii C, XWY iii PQR iv Vrious nswers possile. 12 If the sum of the two ngles is less thn 90. If the sum of the two ngles is equl to 90. If the sum of the two ngles is greter thn 90. d This sttement is untrue s two ute ngles n t dd to more thn Yes Right ngle N V Stright ngle Reflex ngle Otuse ngle ute ngle Topi 5 Geometry 195

53 .s prt. mesurement ND geometry Chllenge Tringles 1 ΔC Slene ΔLMN Equilterl ΔXYZ Isoseles d ΔPQR Isoseles e ΔRST Slene f ΔEFG Isoseles 2 Right-ngled tringle ute-ngled tringle ute-ngled tringle d ute-ngled tringle e Otuse-ngled tringle f Right-ngled tringle 3 S V e R P C Q T R 4 5 D 6 x = 57 g = 125 t = 44 d k = 60 e f = 93 f z = 90 7 d = 45 = 65 = 50 8 = 70 e = 76 n = 52 9 u = 66 k = 45 d = 76 d t = 74 e f = 57 f p = p = 50 ; n = 70 = 22 ; = 28 x = 80 ; y = 50 d t = 35 ; s = 55 e = 46 f n = 21 g m = 60 h t = Isoseles/ute-ngled tringle 12 Chek with your teher. 13 Chek with your teher m (,, d; hek with your teher) 16 = 20 p = 60 k = 30 d p = d Chek with your teher. 19 Yes, ngles dd to 180. No, ngles dd to 175. No, ngles dd to 185. d Yes, ngles dd to The sum of the ngles will e greter thn slene tringles, 6 equilterl tringles nd 24 isoseles tringles Yes d The lrgest ngle is lwys opposite the longest side (likewise, the smllest ngle is opposite the shortest side nd the middle size ngle is opposite the middle length side). lso, ngles tht re opposite to sides of equl length re equl in size. e The sum of ny two side lengths must e greter thn the length of the third side. d f U M M N P W N O 22 C C 40 d C e 20 f C D E D E F g 10 h The ngle is hlved every time new tringle is onstruted. 23 MPN = 72 nd NMP = 36 MPN = 54 nd MNP = Use protrtor to mesure the three ngles of the tringle, nd reord these vlues in tle like the one shown. Tringle C Type of tringle Right-ngled 2 90 Vrious vlues possile D Vrious vlues possile Right-ngled Vrious possile digrms s prt. d s prt. e s prt. f s prt. g ny tringle formed with one of its sides s the dimeter of the irle nd the three verties on the irle is right-ngled tringle. 196 Mths Quest 7

54 ` mesurement nd geometry Chllenge Qudrilterls nd their properties 1 Squre Trpezium Kite d Rhomus e Retngle f Irregulr qudrilterl 2 E E 3 t = 138 = 85 t = 90 d t = 202 e p = 92 f m = 90 g s = 53 h = 66 i k = 67 4 m = 78 (opposite ngles of prllelogrm equl) u = 75 (opposite ngles of rhomus equl) t = 132 (opposite ngles of kite equl) d f = 108 (opposite ngles of kite equl) e p = 63 (opposite ngles of rhomus equl) f z = 73 (opposite ngles of prllelogrm equl) 5 m = 113 = 98 ; d = 82 t = 116 d p = 52 ; m = 106 e t = 103 ; x = 91 f m = 65 ; p = 115 ; n = 65 6 D 7 E 8 True True Flse d True e True 9 D Chek with your teher. 13 p = nd Yes, ngles dd up to 360. No, ngles dd up to Infinite numer Vrious nswers 19 Two smller otuse-ngled isoseles tringles; one lrger size otuse-ngled isoseles tringle nd two right-ngled slene tringles. Two types: retngles (of different sizes) nd rhomus. 8 m m 60 m d In oth right-ngled tringles ngles re: 27, 63 nd 90 ; in ll isoseles tringles ngles re: 126, 27, 27 ; in rhomus ngles re: 126, 54, 126, 54 ; in ll retngles ll ngles re 90 eh. e Vrious nswers 20 Consider the sum of the ngles of the two tringles C nd CD. In ΔC: n + p + q = 180 In ΔCD: m + r + s = 180 In qudrilterl CD: n + p + q + m + r + s = Prllel nd perpendiulr lines 1 Digrm Type of ngle ssoited shpe Rule Corresponding F re equl lternte Z re equl Co-interior C dd to C D d 4 nd d, nd, e nd h, f nd g nd e, nd g, nd f, d nd h nd e, d nd f d nd f, d nd e 5 C D d e D 6 i Corresponding ngles ii p = 60 i lternte ngles ii q = 45 i Vertilly opposite ngles ii s = 65 d i Corresponding ngles ii t = 72 e i Co-interior ngles ii m = 110 f i Co-interior ngles ii n = 48 7 y = 116, orresponding z = 38, lternte = 136, supplementry Topi 5 Geometry 197

55 mesurement ND geometry d g = 68, vertilly opposite e h = 90, orresponding f k = 120, lternte 8 i = 123 ii x = 43 iii y = 28 iv z = 50 v p = 70 vi q = 45 In iii, H GC In v, PW VR In vi, NR QV, MS VQ 9 x = y = No, s For prllel lines, o-interior ngles need to dd to 180 (or lternte ngles must e equl). 14 Chek with teher. 15 ute PQR = 67 nd reflex PQR = The vertil lines re prllel, the horizontl lines re prllel nd the olique lines re prllel euse they re eqully distnt from eh other. The vertil lines re prllel nd the horizontl lines re prllel euse they re eqully distnt from eh other. Investigtion Rih tsk 1 2 d 3 4 Code puzzle Jpn Yen, US Dollr, Itly Lire, Indi Rupee, Frne Frn, Englnd Pound, Spin Pesets, New Guine Kin. 198 Mths Quest 7

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