STRAND F: GEOMETRY F1 Angles and Symmetry

Transcription

1 PRIMRY Mthemtis SKE, Strn F UNIT F1 ngles n Symmetry: Text STRND F: GEOMETRY F1 ngles n Symmetry Text ntents Setin F1.1 Mesuring ngles F1.2 Line n Rttinl Symmetry F1.3 ngle Gemetry F1.4 ngles with Prllel n Interseting Lines F1.5 ngle Symmetry in Regulr Plygns IMT, Plymuth University

2 0 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text F1 ngles n Symmetry F1.1 Mesuring ngles prtrtr n e use t mesure r rw ngles. Nte The ngle run mplete irle is 360. The ngle run pint n stright line is Wrke Exmple 1 Mesure the ngle in the tringle shwn. Slutin Ple prtrtr n the tringle s shwn. The ngle is mesure s Nte When mesuring n ngle, strt frm the 0 whih is in line with n rm f the ngle. IMT, Plymuth University 1

3 F1.1 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text Wrke Exmple 2 Mesure the mrke ngle. Slutin Using prtrtr, the smller ngle is mesure s 100. S require ngle = = Wrke Exmple 3 Drw ngles f () 120 () 330. Slutin () Drw hrizntl line. Ple prtrtr n tp f the line n rw mrk t Then remve the prtrtr n rw the ngle. 120 () T rw the ngle f 330, first sutrt 330 frm 360 : = 30 Drw n ngle f The lrger ngle will e IMT, Plymuth University 2

4 F1.1 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text Exerises 1. Estimte the size f eh ngle, then mesure it with prtrtr. () () () () (e) (f) 2. Drw ngles with the fllwing sizes. () 50 () 70 () 82 () 42 (e) 80 (f) 100 (g) 140 (h) 175 (i) Mesure these ngles. () () () () (e) (f) 4. Drw ngles with the fllwing sizes. () 320 () 190 () 260 () 210 (e) 345 (f) Mesure eh nme (,, ) ngle elw n up the ngles in eh igrm. Wht yu ntie? () () IMT, Plymuth University 3

5 F1.1 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text () () 6. Fr eh tringle elw, mesure eh interir ngle n up the three ngles yu tin. () () () IMT, Plymuth University 4

6 F1.1 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text () D yu tin the sme finl result in eh se? 7. In eh igrm elw, mesure the ngles mrke with letters n fin their ttl. Wht yu ntie ut the ttls? () () () () 8. () Drw stright line JK tht is 10 m lng. L () () Drw ngles f 40 n 50 t J n K respetively, t frm the tringle JKL shwn in the igrm. J m 50 Mesure the lengths f JL n KL n the size f the remining ngle. T IMT, Plymuth University 5

7 F1.1 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 9. The igrm shws rugh sketh f qurilterl. D () () Drw the qurilterl urtely. Mesure the length f D 4 m n the size f the ther tw ngles m m 10. Mesure the interir (insie) ngles f these qurilterls. In eh se fin the ttl sum f the ngles. Wht yu ntie? () () 11. Drw tw ifferent pentgns. () Mesure eh f the ngles in th pentgns. () up yur nswers t fin the ttl f the ngles in eh pentgn. () D yu think tht the ngles in pentgn will lwys up t the sme numer? F1.2 Line n Rttinl Symmetry n jet hs rttinl symmetry if it n e rtte ut pint s tht it fits n tp f itself withut mpleting full turn. The shpes elw hve rttinl symmetry. In mplete turn this shpe In mplete turn this shpe fits n tp f itself tw times. fits n tp f itself fur times. It hs rttinl symmetry f rer 2. It hs rttinl symmetry f rer 4. IMT, Plymuth University 6

8 F1.2 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text Shpes hve line symmetry if mirrr ul e ple s tht ne sie is n ext refletin f the ther. These imginry 'mirrr lines' re shwn y tte lines in the igrms elw. This shpe hs This shpe hs 2 lines f symmetry. 4 lines f symmetry. Wrke Exmple 1 Fr the given shpe, stte: () the numer f lines f symmetry, () the rer f rttinl symmetry. Slutin () There re 3 lines f symmetry s shwn. () There is rttinl symmetry with rer 3, euse the pint mrke ul e rtte t ' then t '' n fit extly ver its riginl shpe t eh f these pints. Exerises 1. Whih f the shpes elw hve () line symmetry () rttinl symmetry? Fr line symmetry, py the shpe n rw in the mirrr lines. Fr rttinl symmetry stte the rer. IMT, Plymuth University 7

9 F1.2 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text D E F G H I 2. Fr eh shpe elw stte: () whether the shpe hs ny symmetry; () hw mny lines f symmetry it hs; () the rer f symmetry if it hs rttinl symmetry. 3. py n mplete eh shpe elw s tht it hs line symmetry ut nt rttinl symmetry. Mrk lerly the lines f symmetry. () () () IMT, Plymuth University 8

10 F1.2 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text () (e) (f) 4. py n, if pssile, mplete eh shpe elw, s tht they hve rttinl symmetry, ut nt line symmetry. In eh se stte the rer f the rttinl symmetry. () () () () (e) (f) 5. py n mplete eh f the fllwing shpes, s tht they hve th rttinl n line symmetry. In eh se rw the lines f symmetry n stte the rer f the rttinl symmetry. () () () () (e) (f) 6. Drw squre n shw ll its lines f symmetry. 7. () Drw tringle with: (i) 1 line f symmetry (ii) 3 lines f symmetry. () Is it pssile t rw tringle with 2 lines f symmetry? 8. Drw shpe whih hs 4 lines f symmetry. IMT, Plymuth University 9

11 F1.2 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 9. Drw shpe with rttinl symmetry f rer: () 2 () 3 () 4 () n yu rw: () pentgn with extly 2 lines f symmetry, () hexgn with extly 2 lines f symmetry, () n tgn with extly 3 lines f symmetry? 11. These re the initils f the Interntinl ssitin f Whistlers. I W Whih f these letters hs rttinl symmetry? 12. Whih f the esigns elw hve line symmetry? () () Tj Mhl flr tile sin rpet esign () () (e) ntemprry rt Wllpper pttern Tile esign 13. () py n rw the refletin f this shpe in the mirrr line. IMT, Plymuth University 10

12 F1.2 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text () py n mplete the igrm ppsite s tht it hs rttinl symmetry. () Wht is the rer f rttinl symmetry f this shpe? F1.3 ngle Gemetry There re numer f imprtnt results nerning ngles in ifferent shpes, t pint n n line. In this setin the fllwing results will e use. 1. ngles t Pint The ngles t pint will lwys up t 360. It es nt mtter hw mny ngles re frme t the pint their ttl will lwys e = ngles n Line ny ngles tht frm stright line up t ngles in Tringle The ngles in ny tringle up t = = ngles in n Equilterl Tringle In n equilterl tringle ll the ngles re 60 n ll the sies re the sme length ngles in n Isseles Tringle In n isseles tringle tw sies re the sme length n tw ngles re the sme size. 6. ngles in Qurilterl The ngles in ny qurilterl up t 360. equl ngles = 360 IMT, Plymuth University 11

13 F1.3 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text Wrke Exmple 1 Fin the sizes f ngles n in the igrm elw Slutin First nsier the qurilterl. ll the ngles f this shpe must up t 360, s = = 360 = = 100 Then nsier the stright line frme y the ngles n. These tw ngles must up t 180 s, ut = 100, s + = = 180 = = 80 Wrke Exmple 2 40 Fin the ngles,, n in the igrm Slutin First nsier the tringle shwn. The ngles f this tringle must up t 180, 40 S, = IMT, Plymuth University 12

14 F1.3 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text Next nsier the ngles run the pint shwn. The three ngles must up t 360, s ut = 110, s = = = 360 = = 130 Finlly, nsier the sen tringle. The ngles must up t 180, s + + = 180 s this is n isseles tringle the tw ngles, n, must e equl, s using = n the ft tht = 130, gives s = 25, = = = = 50 = 25 Wrke Exmple 3 In the figure elw, nt rwn t sle, is n isseles tringle with = p n = ( p + 3 ). p (p+3) () Write n expressin in terms f p fr the vlue f the ngle t. () Determine the size f EH ngle in the tringle. IMT, Plymuth University 13

15 F1.3 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text Slutin () s is n isseles tringle, = p + 3 () Fr tringle, ( ) + ( + ) = p + p + 3 p p + 6 = 180 (tke 6 frm eh sie) 3p = p = 174 (ivie th sies y 3) p = 58 Exerises 1. Fin the size f the ngles mrke with letter in eh igrm. () () () x () (e) (f) x (g) (h) (i) 70 x x (j) (k) (l) IMT, Plymuth University 14

16 F1.3 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text (m) (n) () x x () Fr eh tringle, fin the ngles mrke n. (i) (ii) (iii) () () Wht yu ntie ut the ngle mrke n the ther tw ngles given in eh prlem? Fin the size f the ngle in eh prlem elw withut wrking ut the size f ny ther ngles. (i) (ii) (iii) The igrm elw shws retngle with its ignls rwn in. 22 () py the igrm n mrk in ll the ther ngles tht re 22. () Fin the sizes f ll the ther ngles. IMT, Plymuth University 15

17 F1.3 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 4. Fin the ngles mrke with letters in eh f the fllwing igrms. In eh igrm the lines ll lie insie retngle. () () f e 15 g () 10 () e 80 e f 5. Fin the ngles mrke with letters in eh qurilterl elw. () () () () e e 50 IMT, Plymuth University 16

18 F1.3 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text (e) (f) h g f e 22 g e h f D i 80 is stright line. 6. swing is uilt frm tw metl frmes. sie view f the swing is shwn elw. e E f 68 D The lengths f n E f the swing re the sme n the lengths f n D f the swing re the sme. Fin the sizes f the ngles,,,, e n f. 7. The igrm shws wen frme tht frms prt f the rf f huse. f 45 e Fin the sizes f the ngles,,,, e n f. IMT, Plymuth University 17

19 F1.3 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 8. The igrm shws the pln fr nservtry. Lines re rwn frm the pint O t eh f the ther rners. Fin ll the ngles mrke with letters, if O 20 g 20 f ˆ = D ˆ = DE ˆ = 135 E e 135 D 9. Write wn n equtin n use it t fin the vlue f x in eh igrm. () () () 2x 4x x 20 3x x + 20 x 20 x +10 x +10 x () (e) (f) x 5 x +10 2x x x x x +10 x 2x + 10 x +15 x 20 (g) (h) (i) 3x 5x x 4x 150 2x 3x x x (j) (k) (l) 4x 4x 10 2x 10 8x 5x 80 6x IMT, Plymuth University 18

20 F1.3 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 10. The igrm shws regulr hexgn. O is the pint t the entre f the hexgn. n re tw verties. O () Write wn the rer f rttinl symmetry f the regulr hexgn. () Drw the lines frm O t n frm O t. (i) Write wn the size f ngle O. (ii) Write wn the mthemtil nme fr tringle O. 11. lulte ngles D n, giving resns fr yur nswers Nt t sle F1.4 ngles with Prllel n Interseting Lines D Oppsite ngles When ny tw lines interset, tw pirs f equl ngles re frme. The tw ngles mrke re pir f ppsite equl ngles. The ngles mrke re ls pir f ppsite equl ngles. rrespning ngles When line intersets pir f prllel lines, =. The ngles n re lle rrespning ngles. lternte ngles The ngles n re equl. Prf This result fllws sine n e re ppsite ngles, s = e, n e n re rrespning ngles, s =. Hene = e = The ngles n re lle lternte ngles. e Supplementry ngles The ngles n up t 180. Prf This result fllws sine + = 180 (stright line), n = sine they re rrespning ngles. Hene + = 180. These ngles re lle supplementry ngles. IMT, Plymuth University 19

21 F1.4 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text Wrke Exmple 1 Fin the ngles mrke, n. Slutin There re tw pirs f ppsite ngles here s: = 100 n = 100 ls n frm stright line s Wrke Exmple 2 + = = 180 = 80, s = 80 Fin the sizes f the ngles mrke,, n in the igrm. Slutin First nte the tw prllel lines mrke with rrw hes. 70 Then fin. The ngle n the ngle mrke 70 re ppsite ngles, s = 70. The ngles n re lternte ngles s = = 70. The ngles n re ppsite ngles s = = 70. The ngles n re pir f interir ngles, s + = 180, ut = 70, s 70 + = 180 = = 110 Wrke Exmple 3 Fin the ngles mrke,, n in the igrm. Slutin T fin the ngle, nsier the three ngles tht frm stright line. S = 180 = = The ngle mrke is ppsite the ngle, s Nw n n e fun using rrespning ngles. = = 50. The ngle n the 70 ngle re rrespning ngles, s = 70. The ngle n the 60 ngle re rrespning ngles, s = 60. IMT, Plymuth University 20

22 F1.4 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text Wrke Exmple 4 E F I 95 z K L J 50 y N x M D G H In the igrm ve, nt rwn t sle, is prllel t D n EG is prllel t FH, ngle IJL = 50 n ngle KIJ = 95. lulte the vlues f x, y n z, shwing lerly the steps in yur lultins. Slutin Vlue f x ngles IG n END re supplementry ngles, s 95 +END ˆ = 180 END ˆ = i.e. END ˆ = 85 ut ngles END n FMD re rrespning ngles, s Vlue f y 85 = x ngles D (y) n re lternte ngles, s In tringle IJ, y = ˆ y = 180 i.e. y = 35 y = 180 ( ) = Vlue f z ngles KH ˆ (z) n FMD (x) re lternte ngles, s z = x i.e. z = 85 IMT, Plymuth University 21

23 F1.4 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text Exerises 1. Fin the ngles mrke in eh igrm, giving resns fr yur nswers. () () () () 80 (e) 35 (f) 50 (g) (h) (i) (j) (k) (l) (m) (n) () IMT, Plymuth University 22

24 F1.4 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 2. Fin the size f the ngles mrke,,, et. in eh f the igrms elw. () 110 () () () (e) 40 (f) e (g) (h) 42 e f (i) (j) IMT, Plymuth University 23

25 F1.4 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 3. y nsiering eh igrm, write wn n equtin n fin the vlue f x. () () x 3x 2x 3x 3x 2x () () 5x 6x 3x 3x (e) (f) 4x 3x 5x 2x 4. Whih f the lines shwn elw re prllel? E H 66 G 66 J I L K D F IMT, Plymuth University 24

26 F1.4 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 5. The igrm shws the pth f pl ll s it unes ff ushins n ppsite sies f pl tle () Fin the ngles n. () If, fter the sen une, the pth is prllel t the pth efre the first une, fin n. 6. wrkenh is stning n hrizntl flr. The sie f the wrkenh is shwn. 50 E D The legs n D re equl in length n jine t E. E = E () Whih tw lines re prllel? ngle D is 50. () Wrk ut the size f ngle giving resn fr yur nswer. 7. Here re the nmes f sme qurilterls. Squre Retngle Rhmus Prllelgrm Trpezium Kite () () Write wn the nmes f the qurilterls whih hve tw pirs f prllel sies. Write wn the nmes f the qurilterls whih must hve tw pirs f equl ppsite sies. IMT, Plymuth University 25

27 F1.4 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 8. WXYZ is retngle. W 36 X Nt t sle Z Y () ngle XWY = 36. Wrk ut the size f ngle WYZ, giving resn fr yur nswer. PQRS is rhmus. P 36 Q Nt t sle O () ngle QPR = 36. S R The ignls PR n QS interset t O. Wrk ut the size f ngle PQS, giving resn fr yur nswer. 9. In the igrm, XY = ZY n ZY is prllel t XW. Y q W Nt t sle Z 48 p r X () Write wn the size f ngle p. () lulte the size f ngle q. Give resn fr yur nswer. () Give resn why ngle q = ngle r. IMT, Plymuth University 26

28 F1.4 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 10. In the igrm shwn elw, DE is pentgn. E = 108, = 90, ED = 80, D = 57 n E is prllel t D. 108 y E 80 x 57 D lulte the size f the ngle mrke () x () y. F1.5 ngle Symmetry in Regulr Plygns Regulr plygns will hve th line n rttinl symmetry. This symmetry n e use t fin the interir ngles f regulr plygn. Wrke Exmple 1 Fin the interir ngle f regulr egn. Slutin The igrm shws hw regulr egn n e split int 12 isseles tringles. s there re 360 run the entre f the egn, the entre ngle in eh tringle is 360 = Interir ngles 30 S the ther ngles f eh tringle will tgether e Therefre eh f the ther ngles will e = = 75 2 IMT, Plymuth University 27

29 F1.5 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text s tw jent ngles re require t frm eh interir ngle f the egn, eh interir ngle will e 75 2 = 150 s there re 12 interir ngles, the sum f these ngles will e = Wrke Exmple 2 Fin the sum f the interir ngles f regulr heptgn. Slutin Split the heptgn int 7 isseles tringles. Eh tringle ntins three ngles whih up t 180, s the ttl f ll the mrke ngles will e = Hwever the ngles t the pint where ll the tringles meet shul nt e inlue, s the sum f the interir ngles is given y Wrke Exmple 3 () () = 900 py the tgn shwn in the igrm n rw in ny lines f symmetry. py the tgn n she in extr tringles s tht it nw hs rttinl symmetry. Slutin () There is nly ne line f symmetry s shwn in the igrm. F G E D () The riginl tgn hs n rttinl symmetry. y shing the extr tringle shwn, y shing ll the tringles, it hs it hs rttinl symmetry f rer 4. rttinl symmetry f rer 8. IMT, Plymuth University 28

30 F1.5 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text Exerises 1. Fin the interir ngle fr regulr: () pentgn () hexgn () tgn () egn (10 sies). 2. Fin the sum f the interir ngles in eh plygn shwn elw. () () 3. Whih regulr plygns hve interir ngles f: () 90 () 120 () 108 () 140 (e) 60 (f) 144? 4. Mke 3 pies f eh shpe elw. n she prts f them, s tht: () () () they hve line symmetry, ut n rttinl symmetry; they hve line symmetry n rttinl symmetry; they hve rttinl symmetry, ut n line symmetry. In eh se rw in the lines f symmetry n stte the rer f rttinl symmetry. 5. () Drw shpe tht hs rttinl symmetry f rer 3 ut n line symmetry. () Drw shpe tht hs rttinl symmetry f rer 5 ut n line symmetry. 6. () Fr this shpe, is it pssile t she smller tringle s tht is hs rttinl symmetry f (i) 2 (ii) 3 (iii) 4 with n lines f symmetry? IMT, Plymuth University 29

31 F1.5 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text () Is it pssile t she smller tringles s tht the shpe hs (i) 1 (ii) 2 (iii) 3 lines f symmetry n n rttinl symmetry? 7. () plygn hs 9 sies. Wht is the sum f the interir ngles? () py n mplete the tle elw. Shpe Sum f interir ngles Tringle 180 Squre Pentgn Hexgn 720 Heptgn Otgn () () (e) Desrie rule tht ul e use t lulte the sum f the interir ngles fr plygn with n sies. Fin the sum f the interir ngles fr 14-sie plygn. The sum f the interir ngles f plygn is Hw mny sies es the plygn hve? 8. () regulr plygn with n sies is split int isseles tringles s shwn in the igrm. Fin frmul fr the size f the ngle mrke θ. () Use yur nswer t prt () t fin frmul fr the interir ngle f regulr plygn with n sies. θ () Use yur frmul t fin the interir ngle f plygn with 20 sies. 9. () Write wn the rer f rttinl symmetry f this retngle. () Drw shpe whih hs rttinl symmetry f rer 3. () (i) Hw mny lines f symmetry hs regulr pentgn? (ii) Wht is the size f ne exterir ngle f regulr pentgn? IMT, Plymuth University 30

32 F1.5 PRIMRY Mthemtis SKE, Strn F1 ngles n Symmetry: Text 10. The piture shws lrge tile with nly prt f its pttern fille in. mplete the piture s tht the tile hs 2 lines f symmetry n rttinl symmetry f rer regulr tgn, rwn ppsite, hs eight sies. One sie f the tgn hs een extene t frm ngle p. q () Wrk ut the size f ngle p. () Wrk ut the size f ngle q. Nt t sle p 12. Q P x T The igrm shws three ientil rhmuses, P, Q n T. () Explin why ngle x is 120. () Rhmus Q n e rtte nt rhmus T. (i) Mrk entre f rttin. (ii) Stte the ngle f rttin. () () Write wn the rer f rttinl symmetry f (i) rhmus (ii) regulr hexgn. The given shpe ul ls represent three imensinl shpe. Wht is this shpe? Investigtin Hw mny squres re there in the given figure? IMT, Plymuth University 31