Angles. Angles. Curriculum Ready.

Transcription

1 ngles ngles urriculum Redy

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3 ngles mesure the mount of turn in degrees etween two lines tht meet t point. Mny gmes re sed on interpreting using ngles such s pool, snooker illirds. lck ll?c White ll?c Write down some other sports/gmes tht you cn think of tht require n understing of ngles: Give this go! While performing circulr llet move, net turned the first hlf esily then with some extr effort, mde it 5 of the remining wy round. ow mny degrees ws net wy from 6 completing the full circle? int: hlf circle is 180 degrees.?c Strt position Work through the ook for gret wy to solve this ngles SRIS 10 TI 1

4 ow does it work? ngles rts of n ngle ngles re formed when two stright rys extend from common point. The mount of rottion swept from one rm to the next in degrees is how they re mesured Rys re stright lines with n rrow on one end only. rm ngle swept y rms in degrees ( c) Vertex rm Rys tht form n ngle re clled rms Nming ngles These two methods of nming use the symol +in front to men 'ngle'. Nme these two ngles: R Method 1 + or + + R or + R The letter t the vertex is lwys written in the middle Method For ngles like these, you cn just use the letter t the vertex If there is more thn one ngle t the sme point, you must use method 1 to reference the ngle properly. Nme these ngles mrked with dot: K L is shred y oth ngles (ommon rm) + or + + KN or + NK N M 2 10 ngles SRIS TI

5 ow does it work? our Turn ngles rts of n ngle 1 ighlight the section of the ngle tht mtches the lel underneth. c RTS F N NGL * RTS F N NGL *.../.../20... rm rm The vertex d e f The ngle swept The rm shred y oth ngles R The ngle swept y the rms 2 Write down the prts of the ngles tht hve een highlighted elow. c d e f M S R L N ngles SRIS 10 TI 3

6 ow does it work? our Turn ngles Nming ngles 1 Nme ech of these ngles. c G NMING NGLS * NMING NGLS *.../.../20... F K 2 Nme ech of the ngles mrked with: dot squre M c N W F 3 Nme the rm common to oth mrked ngles in question 2, (write no rm common if there isn t one). c 4 Nme the ngles indicted in ech of these: c K F G L M = = 4 10 ngles SRIS TI

7 ow does it work? ngles ngle types This tle shows how ngles re clssified y their size. icture Size Nme etween 0c 90c or cute ngle 0c c xctly 90c Smll ox mens 90c or + = 90c Right ngle etween 90c 180c or 90c c tuse ngle xctly 180c Vertex or + = 180c Stright ngle etween 180c 360c or 180c 1 reflex c + could lso e n otuse (or cute) ngle, so include the word reflex in front Reflex ngle xctly 360c Reflex ngle Vertex / or + = 360c or Full rottion ngles SRIS 10 TI 5

8 NGL TS * NGL TS * NGL TS * ow does it work? our Turn ngles ngle types 1 Sketch lel ngles tht mtch ech of these descriptions: tuse ngle + cute + R.../.../20... c Right-ngle + MLN (int: rememer the ox) d Reflex + GU e Full revolution + KL f Stright + F 2 Fill in the tle elow with ll the ngles you cn find mtching the types in the digrm elow: R S U T V W cute ngle Right ngle tuse ngle Stright ngle Reflex ngle + RS + RS + RW + R reflex + RW 6 10 ngles SRIS TI

9 ow does it work? ngles Using protrctor to mesure ngles The mount of turn etween ech rm is mesured in degrees with the id of protrctor. Mesure the size of + Step 1: Set up protrctor to mesure Rememer: the vertex is the pointy it lce the centre mrker on the protrctor t the vertex Line up one of the rms with 0cmrk Step 2: Red the ngle Mesure with the outside scle s it strts with 0c ` + = 120c The two scles on protrctor enle us to mesure ngles from either direction. Mesure the size of + Step 1: Set up protrctor to mesure Line up one of the rms with 0cmrk lce the centre mrker on the protrctor t the vertex Step 2: Red the ngle Mesure with the inside scle s it strts with 0c ` + = 65c ngles SRIS 10 TI 7

10 ow does it work? our Turn ngles Using protrctor to mesure ngles 1 Write down the size (mount of turn in degrees) of these mesured ngles. R ` + = ` + R = c d K L ` + LK = ` + = 2 Write down the size of the ngles indicted elow ech digrm. ` + = ` + = c d ` + = ` + = 8 10 ngles SRIS TI

11 USING RTRTR T MSUR NGLS * ow does it work? our Turn ngles Using protrctor to mesure ngles 3 Try these trickier ones! R S.../.../20... T ` + RS = ` + = 4 Use protrctor to mesure the size (mount of turn in degrees) etween the rms for these four ngles: ` + = ` + = c d ` + = ` + = ngles SRIS 10 TI 9

12 ow does it work? our Turn ngles Using protrctor to mesure ngles 5 Mesure ech cute ngle etween the stright supports on ert the spider s we mtch the letter with the correct size elow. T L U I R S N V 44c 10c 24c 30c 20c 40c 35c 52c 27c 22c 56c ngles SRIS TI

13 ow does it work? ngles Using protrctor to mesure reflex ngles Most protrctors only mesure ngles up to 180c, so mesure the ngle you cn go from there. Mesure the size of reflex + R Step 1: Mesure the otuse + R tuse + R = 140c Step 2: Sutrct the size of the otuse ngle from 360c 140c R tuse + R + reflex + R = 360c ( full revolution) R ` Reflex + R = 360c - 140c = 220c ere is nother exmple with n cute ngle. Mesure the size of reflex Step 1: Mesure the cute + F + F Step 2: Sutrct the size of the cute ngle from 360c F 25c tuse + F + reflex + F = 360c ( full revolution) F ` Reflex + F = 360c - 25c = 335c ngles SRIS 10 TI 11

14 USING RTRTR T MSUR RFL NGLS * ow does it work? our Turn ngles Using protrctor to mesure reflex ngles 1 lculte the size of these reflex ngles. L K ` Reflex + = 360c - = ` Reflex + LK = 360c - = c d T U V ` Reflex + = 360c - = ` Reflex + TUV = 360c - = 2 Mesure write down the size of the reflex ngle for ech of these: M L ` Reflex + LMN = N c.../.../20... I ` Reflex + = ` Reflex + I = ngles SRIS TI

15 Where does it work? ngles djcent ngles ngles tht do not overlp shre n rm from the sme vertex point re clled djcent ngles. Nme the djcent cute ngles in this digrm rm is shred y + + Sme vertex point for is djcent to + djcent simply mens 'next to' ere is the officil wy to sy it: The rm is common to oth ngles. The vertex is common to oth ngles. ere is n exmple where ngles with common rm vertex overlp. Nme ll the otuse ngles djcent to + U V W U rm U is lso common to these otuse ngles + UW + U The vertex is common to ll ngles rm is lso common to these otuse ngles: + V, + + W + U is djcent to the otuse ngles + UW, + U, + + W. e creful: g: + V + U shre common rm vertex, ut they re not djcent ecuse they overlp. U V rm vertex common to oth ngles SRIS 10 TI 13

16 NT NGLS * NT NGLS * Where does it work? our Turn ngles djcent ngles 1 Nme pir of djcent cute ngles in ech of these digrms: K L.../.../20... N M 2 Nme one reflex ngle ll the cute ngles djcent to these ngles mrked with dot. W Rememer to write the word reflex infront of the reflex ngles T (iii) U S (iv) R IG TUMS U IF U FIN LL FUR * 3 rw n otuse ngle lel it + R. rw n cute ngle + S djcent to it. 4 ch of these ngles shre n rm. xplin why they re not djcent to ech other. N M + MN + N MN + N re not djcent ecuse: + + re not djcent ecuse: ngles SRIS TI

17 Where does it work? ngles omplementry supplementry ngles These specil nmes re given to pirs of ngles tht dd together to totl of 90c or 180c. omplementry ngles re pir of ngles tht mke right-ngle (90c) when put together. lculte the size of + if it is the complement of + Rememer like this: omplementry ngles mke orner. 50c Size of = 90c ` c = 90c ` + = 40c If you drw complementry ngles djcent to ech other, you will mke right-ngle! + + re complementry ngles 40c = 50c ere is nother exmple. Nme the pir of complementry ngles in this digrm W 42c 47c V 45c 43c Look for pir of ngles tht dd to 90c U + UV + + = 43c + 47c = 90c ` + UV + re complementry ngles ngles SRIS 10 TI 15

18 Where does it work? ngles Supplementry ngles re pir of ngles tht mke stright-ngle (180c) when put together. lculte the supplement of 132c The supplement of 132c is: 180c - 132c = 48c Supplementry ngles dd to 180c lculte the size of n ngle supplementry to + W S Rememer like this: Supplementry ngles mke Stright ngle. W 44c 65c + W = + W + + = 65c + 44c = 109c + W is formed y two djcent ngles + W + ngle = 180c ` 109c + ngle = 180c ` ngle = 180c - 109c ` ngle = 71c ` the size of the ngle supplementry to +W is 71c uestions with mny ngles need closer investigtion. Write down the pir of djcent, supplementry ngles from this digrm igrm not drwn to scle K 29c 105c 19c 46c L M N + M = + MN + + N + M = 46c + 105c + M = 151c The totl size of other djcent ngles is sometimes needed + M + + K = 151c + 29c = 180c + M + K re supplementry ngles ngles SRIS TI

19 MLMNTR N SULMNTR NGLS * Where does it work? our Turn ngles omplementry supplementry ngles 1 lculte the complement (the ngle tht mkes it 90c) of these ngles: 30c 80c c 46c.../.../20... d 11c e 23. 5c f 18. 3c 2 lculte the supplementry (the ngle tht mkes it 180c) of these ngles: 100c 90c c 165c d 109c e c f c 3 lculte the size of the missing complementry ngles elow: 11. 5c R S T 71c U + = + TSU = 4 lculte the size of the missing supplementry ngles elow: I W 107c F + = + I = ngles SRIS 10 TI 17

20 Where does it work? our Turn ngles omplementry supplementry ngles 5 Nme the pir of supplementry ngles in this digrm: int: wht is the size of + W W V 64c 44c 46c U re supplementry ngles 6 Nme the two pirs of complementry ngles in this digrm: 24c 35c 23c 30c F First pir: 37c + F Second pir: + FG 31c G It's como time! 7 Nme the pir of djcent complementry ngles in this digrm: R S.../.../ c 25c 65c 35c T M TIM * M TIM * M TIM * 25c U V If + UV is drwn djcent to + TU s shown, wht size must it e to mke + V stright ngle? int: the ngles must ll dd to 180c + UV = ngles SRIS TI

21 Where does it work? ngles Verticlly opposite ngles When two stright lines cross ech other, four ngles re creted If you mesured ech of these ngles with your protrctor, you will discover tht: ngle 1 = ngle 3 ngle 2 = ngle 4 In Mthemtics we cll these equl ngles, verticlly opposite ngles. Nme the pirs of verticlly opposite ngles in this digrm: re two stright lines crossing ech other t ` 1 st pir of verticlly opposite ngles re: + + ` 2 nd pir of verticlly opposite ngles re: + + djcent ngles formed y the intersection of two stright lines re supplementry. lculte the size of these ngles: W 140c + + = + W Verticlly opposite ngles re the sme size ` + = 140c +W + W is djcent to + W ` + W + + W = 180c djcent ngles of intersecting lines re supplementry ` + W + 140c = 180c ` + W = 40c ngles SRIS 10 TI 19

22 VRTILL SIT NGLS * VRTILL SIT NGLS * Where does it work? our Turn ngles Verticlly opposite ngles 1 Nme shde ll the pirs of verticlly opposite ngles elow: R M.../.../20... T S L K + TR + TS First pir: Second pir: 2 lculte the size of these ngles: I + I + + G 67c + 49c 56c re stright lines 3 This digrm is mde up of four stright lines,, F G intersecting t the sme point. Nme ten different pirs of verticlly opposite ngles. G (iii) (iv) (v) (vi) F (vii) (viii) (ix) (x) ngles SRIS TI

23 Where does it work? ngles rllel lines rllel lines never cross ech other, so on their own they never form n ngle. Used to show tht the lines re prllel to ech other rllel lines re nmed like this: ;; This symol mens 'is prllel to' We get ngles formed when nother line tht is not prllel crosses them. line tht crosses prllel lines is clled trnsversl The opposite of prllel is perpendiculr. The symol for this is =. It mens the lines cross t 90cto ech other ngles ngles tht re on lternte sides of the trnsversl inside pir of prllel lines re the sme size. G G F F + GF = + FG + FG = + GF These re clled lternte ngles, they form zigzg shpe when highlighted. Let s cll them ngles. Find the size of + M 124c K L + = 124c + = + M ` + M = 124c N M lternte ngles in prllel lines, K ;; LM lternte ngles in prllel lines re the sme size ngles SRIS 10 TI 21

24 Where does it work? ngles Fngles ngles tht re in corresponding (mtching) positions on pir of prllel lines re the sme size. G G F F + F = + FG + FG = + G These re clled corresponding ngles they form n F shpe when highlighted. Let s cll them Fngles. Find the size of + LN G 124c L + = 124c + = + LN ` + LN = 124c N K M orresponding ngles in prllel lines, K ;; LM orresponding ngles in prllel lines re the sme size ngles ngles on the sme side of the trnsversl inside pir of prllel lines re supplementry. G G F + FG + + FG = 180c F + GF + + FG = 180c These re clled cointerior ngles they form shpe when highlighted. Let s cll these ngles. Find the size of + L 124c L + = 124c L = 180c ` + L = 56c N K M ointerior ngles in prllel lines, K ;; LM ointerior ngles in prllel lines re supplementry ngles SRIS TI

25 Where does it work? our Turn ngles rllel lines 1 For ech of these digrms: W Nme the trnsversl. Nme the pir of prllel lines using the correct symol. F G RLLL LINS * RLLL LINS *.../.../ Nme ll the pirs of ngles, Fngles ngles in these digrms: W S T U V ngles (lternte ngles) F G ngles (lternte ngles) Fngles (corresponding ngles) There re four pirs of this type Fngles (corresponding ngles) There re four pirs of this type (iii) (iii) (iv) (iv) ngles (cointerior ngles) ngles (cointerior ngles) ngles SRIS 10 TI 23

26 Where does it work? our Turn ngles rllel lines 3 stright cle N, runs underneth rilwy trck s shown. Use the ngles Fngles properties to complete the tle with ll the other ngles tht re the sme size s the two given. 63c 117c 63c Rememer you + + cn look for L 117c verticlly opposite ngles too s they re lso equl. K N M 4 Find the size of ech of these ngles include one of the properties elow you used to find them: roperties: lternte, corresponding, cointerior, verticlly opposite, stright F 46c G + G = roperty used: orresponding ngles + GF = roperty used: ngles + F = roperty used: ngles + GF = roperty used: ngles Try this one with 3 prllel lines! (psst! ou will need to use more thn one property) V + W = roperty used: W 128c orresponding ngle to + T, ( + ) = 128c + W is verticlly opposite to + = 128c + W = roperty used: T 128c U + = roperty used: ngles SRIS TI

27 Where does it work? our Turn ngles rllel lines Since the rules for ngles, Fngles ngles only work when lines re prllel, you cn use them to find out whether pir of lines re prllel or not! 5 For ech of these: circle prllel or not prllel for the lines drwn write reson why you circled the one you did! The line re: F 153c Reson: RLLL NT RLLL 37c The cointerior ngles do not dd up to 180c or The cointerior ngles re not supplementry S 81c T The line ST UV re: Reson: RLLL NT RLLL U 81c V c R The line RS re: N 101c 100c Reson: RLLL NT RLLL S d N K The line LM N re: L Reson: RLLL NT RLLL (write ll the properties used here) 158c I 22c M ngles SRIS 10 TI 25

28 Wht else cn you do? ngles ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. ngles tht form stright line dd to 180c. lculte the size of + MN, if L is stright line M N + L = 180c( stright ngle) 41c 73c L + LM + + N + + MN = + L = 180c ` 41c + 73c + + MN = 180c ` + MN = 180c - 41c - 73c + MN = 66c ngles tht re prt of full revolution re clled ngles t point they dd to 360c. lculte the size of + ere is right-ngle exmple. lculte the size of = 360c ` 38c + 62c + 125c + + = 360c ` + = 360c - 38c - 62c - 125c + = 135c + KN, + MKN + LKM if they re ll the sme size K M L + KN + + MKN + + LKM = 90c ` + KN, + MKN + LKM = 90c ' 3 ` + KN, + MKN + LKM = 30c N 62c 38c 125c The ngles joined t vertex sum to equl 360c + KL is right-ngle which equls 90c ' 3 s they re ll the sme size ngles SRIS TI

29 Wht else cn you do? our Turn ngles ngle sums 1 For ech of these digrms, clculte the size of the missing ngle: K N 13c 29c L M 17c 14c NGL SUMS * NGL SUMS * NGL SUMS * +.../.../ NK = + = c W 72c 76c 203c d Smll dots cn e used to show equl sized ngles R V U T S + = ch ngle = 2 Verticlly opposite ngles cn e used to help find the unknown ngles for these. re stright lines 127c 83c K MN re stright lines M 34c 88c 76c N K + = + M = ngles SRIS 10 TI 27

30 Wht else cn you do? our Turn ngles ngle sums 3 Use the prllel line ngle properties to help find the size of these ngles: W 70c G 47c U + W = 84c 130c V F I + G = + = + GF = + = + G = + W = + GF = omo Time! 4 Give these tricky ones go! ou hve the skills now to use few different ngle properties for ech one. K re stright, prllel lines. re stright lines. 119c re stright, prllel lines. I 46c 141c K + = + FG = int: find F + F first G + = + = + FI = int: find + IF first ngles SRIS TI

31 Wht else cn you do? ngles ngle prolems Mny rel life prolems cn involve the ppliction of the ngle properties covered in this ooklet. This one uses the ngle sum of revolution. Trinity is lindfolded spun round in children s prty gme. If it tkes her equl-sized steps to complete ech circle, how mny degrees does she turn with ech step? 1 complete circle = 1 revolution =360c ` Numer of degrees turned with ech step = 360c ' = 48c ` Trinity spins 48c with ech step she tkes. lwys nswer prolems with sttement ere is nother prolem. Five people were holding lengths of rope ll tied together t the centre. They need to move round until the ngle etween ech rope is the sme. xplin how ech person should move if Kim Rohn must oth sty still. Kim lculte the size ech ngle needs to e. Wei ` 360c ' 5 = 72c ngles t point dd to 360c 92c 37c 100c 44c 87c Rohn Kim Rohn cnnot move. Wei moves 100c - 72c = 28c counter-clockwise Sung-Li ` ngle etween Kim Wei = 72c rin Sung-Li moves 92c - 72c = 20c clockwise ` ngle etween Kim Sung-Li = 72c Sung-Li's movements leve rin 20c + 37c = 57c wy from her. ` rin moves 72c - 57c = 15c counter-clockwise ` ngle etween Sung-Li rin 87c - 15c = 72c rin's movements tke her 15c closer to Rohn ` ngle etween rin Rohn = 87c - 15c = 72c ` ngle etween Wei Rohn = 72c nly ngle left over ngles SRIS 10 TI 29

32 Wht else cn you do? our Turn ngles ngle prolems 1 While performing circulr llet move, net turned the first hlf esily then with some extr effort, mde it 5 of the remining wy round. 6 ow mny degrees ws net wy from completing the full circle?.../.../20... NGL RLMS * NGL RLMS *?c Strt position Rememer me? She immeditely recovers strts her second move fcing where she hd stopped. If she successfully turns nother 180c in the sme direction, how mny degrees wy from the strt position is net now? 2 ert is uilding nother we, this time etween two stright, prllel ems ;; W. is we hs three stright supports:, G. W G 28c 47c K ert wnts to put in nother stright support K tht psses through, strting t (etween W ) finishing t K (etween ). rw in the support K tht mtches ert s wishes. Wht is the size of + if ll the cute ngles ginst the em W re complementry? ngles SRIS TI

33 Wht else cn you do? our Turn ngles ngle prolems 3 toy root is progrmmed to move to ll of the discs shown elow. It strts on disc fcing in the direction of the rrow. When it reches ech disc, the root remins fcing the direction it ws during the previous move. Nme the order of the discs it moves to if it follows these instructions in order: Turn right-ngle clockwise trvel forwrd to the next disc. omplete full revolution then trvel forwrd to the next disc. Turn counter-clockwise 200c trvel forwrd to the next disc. Turn clockwise 270c trvel forwrd to the next disc. Turn clockwise 80c then trvel in reverse (ckwrds) to the next disc. Turn counter-clockwise n cute ngle trvel forwrd to the lst disc. F G isc order: 4 s prt of tresure hunt, prticipnts must complete puzzles to receive the nme of the next destintion. t one stop, the puzzle is this: 27c Step 1: If + is stright ngle, clculte the complement of + dd it to one of the ngles formed when + is divided into nine equl sized prts. Step 2: lculte the size of reflex +, sutrct the vlue of step 1 from it then dd the supplement of + to the nswer. Wht nswer will win you the nme of the next destintion? ngles SRIS 10 TI 31

34 het Sheet ngles ere is summry of the importnt things to rememer for ngles rt of n ngle Nming ngles R rm ngle swept in degrees (c) +R Vertex rm Rys tht form n ngle The letter t the vertex is lwys written in the middle ngle types cute ngle Right ngle tuse ngle Smll ox mens 90c 0c c + = 90c 90c c Stright ngle Reflex ngle Full revolution or Full rottion Vertex Vertex + = 180c 180c 1 reflex c + = 360c / omplementry supplementry ngles omplementry: pir of ngles whose sum = 90c Together they form right-ngle Supplementry: pir of ngles whose sum = 180c Together they form stright-ngle Verticlly opposite ngles ngle 1 = ngle 3 ngle 2 = ngle 4 3 rllel lines ;; : mens the line is prllel to the line rrows indicte prllel lines Trnsversl lternte ngles orresponding ngles ointerior ngles qul on prllel lines qul on prllel lines Supplementry on prllel lines ngles SRIS TI

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36 ngles