2.4 Angles. Establish that for 180 the direction doesn t matter. If you started off facing North, you would be facing South.

Transcription

1 2.4 Agles Agle ca sometimes refer to a corer (vertex) or to the size of the agle at that vertex. I diagrams we ted to use capital letters to represet poits or sizes of agles ad lower case letters to represet legths of lies; e.g., A, Â, ABC, ABC ˆ, A, ABC for agles ad a, AB, for legths. Pupils may also eed to kow the covetioal ways of idicatig equal agles, equal legths of lies ad parallel lies i diagrams. This is sometimes better covered whe dealig with polygos (sectio 2.1). A full-circle protractor (0-360 ) is a lot more coveiet tha a semicircle (0-180 ) oe, ad it s a advatage if there is o area missig i the middle ad there are cotiuous lies goig out from the cetre to the umbers aroud the edge. Pupils also sometimes fid a 360 protractor easier to hold because of the lump at the cetre. The reaso for the two scales (clockwise ad aticlockwise) may eed explaiig. The covetio that aticlockwise rotatios are positive ad clockwise egative is ofte used. Pupils ca aim for a accuracy of 1. A sharp pecil helps. Sometimes lies eed extedig o a drawig to reach the fie scale aroud the edge of the protractor. Agles are revised i sectios 2.5 ad People maths. Review N, E, S, W ( Naughty Elephats Squirt Water!, or equivalet). Everybody stad up. Let s say this way is orth (or work it out or take a vote! West could be the directio of the widows, etc.). Which way is clockwise? Which way is right? Tur 90 clockwise, tur 270 aticlockwise, tur 450 clockwise, etc. Everyoe does it at the same time. The try it metally. Sit dow. If I said tur 270 aticlockwise ad the 90 clockwise which way would you be facig? (N, S, E, W?) How much clockwise is equivalet to x aticlockwise? Give me directios for gettig from here to the hall/diig room/head s office, etc NEED blidfold (clea tea-towels are coveiet), prize (e.g., chocolate bar). Choose 2 voluteers, oe who does t mid beig blidfolded ad blidfold that oe. Escort the blidfolded perso to the back of the room ad rearrage the desks a little. Place the prize somewhere. The blidfolded perso has to get the chocolate bar without touchig aythig except the floor with ay part of him/her. The other voluteer has to give directios but is t allowed aywhere ear the blidfolded perso. Keep close by i case the perso falls. Avoid embarrassig pupils who have difficulty distiguishig right ad left. Right is the had most people write with. The first figer ad thumb of the left had make a L shape whe held out at 90. Establish that for 180 the directio does t matter. If you started off facig North, you would be facig South. Aswer: (360 x), if 0 x 360, or more geerally (360 xmod360). Not allowed to draw aythig or wave arms aroud. Imagie you were usig the telephoe. Obviously check they ca t see aythig. Spi roud to lose orietatio. O a chair tucked uder a table is quite difficult. You could give them 3 lives. Keep it ope-eded ad see what they do geerally they ll use left ad right, but possibly agles, especially ear the ed whe small movemets are ecessary Coverig the special ames for agles i particular rages gives a good opportuity to review > ad < or to itroduce (does ayoe kow Greek?) as a symbol ofte used to represet a agle (a bit like x stadig for a umber i algebra). The words horizotal (the same directio as the horizo) ad vertical might eed revisio. These may be ways of addig iterest to revisio material.

2 acute agles: 0 90 ; right agle (quarter tur): 90 ; obtuse agles: ; straight lie (half tur): 180 ; reflex agles: ; full tur: 360. Why do you thik there are 360 i a full tur? We could measure i % (25% for a quarter tur, etc.) or miutes (15 mi for a quarter tur) Testig Agle Accuracy. Draw axes from 0 to 10 horizotally ad 0 to 30 vertically. Must use the same scale (e.g., 0.5 cm for 1 uit) o both axes. As accurately as possible, joi (0,0) to (10,10). Measure the agle made by this lie ad the horizotal axis. The agle facts that the agles o a straight lie add up to 180 ad the agles at a poit add up to 360 are really just defiitios of what we cout as straight ad what we mea by all the way roud together with our choice of how may degrees to have i a full tur. Just a covetio/historical accidet; apparetly the Babyloias couted i 60 s (base 60) istead of 10 s like we do. We could use 2 (radias) or 400 (gradiets) or aythig you like for a full tur. Pupils may ejoy the chace to press the mysterious ta butto o the calculator before they lear about it i trigoometry. Aswer: Should obviously be 45 because we re bisectig the 90 betwee the axes, but the calculator kows what it should be: make sure you re i 1 degrees mode ad do ta 1. Now joi (0,0) to (10,20) ad agai measure the agle this lie makes with the horizotal axis. Fially (0,0) to (10,30). Pupils may suspect that the lie joiig, say, (0,0) to 1 (10,25) would make a agle of ta 2.5 with the x -axis What is a agle? (Imagie your little brother/sister wated to kow what your maths homework was about.) Tell me a job i which you d have to thik about agles? Parallel ad perpedicular logically crop up here, because parallel lies are straight lies goig i the same directio (agle betwee them = 0 ) ad perpedicular lies are lies that are at 90 to each other. (Eve if the lies go o for ever i both directios, perpedicular lies do t ecessarily touch i 3 dimesios. No-parallel o-itersectig lies are called skew lies.) Pupils may kow from reflectio ad refractio i sciece that ormal meas at 90 (the agles of light rays are measured from lies ormal to the surfaces) Fid out how much the Leaig Tower of Pisa leas (could do this for homework). Check with Check with 1 ta 2 = ta 3 = Further poits may be chose if more practice is eeded; this saves photocopyig sheets of radom agles for pupils to measure, ad reviews coordiates at the same time. It may also be more iterestig. It s quite hard to explai; you ca t really say distace betwee, etc. Need to talk about rotatig or turig: a way of sayig how much somethig has tured. flyig a plae, footballer, plumber, roofer, TV aerial fitter, maths teacher, etc. It is t eough to say that parallel lies are just lies the same distace apart because so are the circumfereces of cocetric circles or railway tracks goig roud a bed. The lies also have to be straight. Could discuss ifiity. Do parallel lies ever meet? Not if they re always a certai (o-zero) distace apart. O a globe parallel lies of logitude meet at the N ad S poles. Aswer: about 10, although it varies year by year as it leas more ad the egieers try to straighte it a little NEED Which Agles are Equal? sheets, pecil crayos, protractors perhaps. This seems to work better tha askig pupils to draw The itetio is that pupils draw a circle (about 1 cm radius) at each of the 15 crossig poits ad the usig two colours shade vertically opposite agles the same colour. Equal agles at differet crossig

3 their ow lies, because uless you re careful there is t room to mark the agles clearly. Have some spare copies because whe you ve goe very wrog it s hard to rescue! How may colours will you eed? Agles associated with parallel lies. Learig the ames ca be tedious. First cocetrate o idetifyig which agles are equal to each other ad which pairs of agles sum to 180 (supplemetary agles). If you have a rectagular whiteboard ad a metre stick (or pole for opeig the widows) you ca easily lay the stick across parallel sides of the board at differet agles to show what equals what; otherwise use a oticeboard or desk. poits should also be coloured the same colour. Aswer: 6, because there are 3 pairs of parallel lies, ad each ca itersect ay of the others ad 3 C 2 = 3. Each itersectio creates 2 differet-sized agles, so altogether there will be 6 differet agles. Vertically opposite agles (like scissors or a pair of pupils rulers, like letter X) othig to do with a vertical directio, but agles opposite at a vertex (= poit). Correspodig agles (i correspodig positios, like Chris ad Katie, both at the ed of a row i the classroom, like letter F). Alterate agles (opposite sides of the lie that goes through the parallel pair of lies, like letter Z). Iterior agles the odd oes out because they re ot equal but they sum to 180 (like letter C). (Ufortuately C-agles are iterior ad the C does t stad for correspodig.) The lie crossig a pair of parallel lies is called a trasversal Clock Agles. If you look at the miute ad hour hads of a clock, there are two agles betwee them. I m oly iterested i the smaller of the two agles. Tell me a time whe the agle is 90. Why is ot exactly correct? Will be more or less tha 90? Tell me aother time whe the agle will be just less tha 90? Whe will it be just more? What will the agle be at 9.30? Work out the (smaller) agle betwee the hads at (Pupils ca do some of this metally.) The sum of the three iterior agles i a triagle = 180. NEED small paper triagles, oe per pupil. Had them out. Who s got a early right-agled triagle? Who s got a early equilateral triagle?, etc. Hold it up. Class experimet betwee us we re tryig all sorts of triagles. Optio 1: You ca tear off the corers ad arrage the pieces ext to oe aother to make a straight lie. Iterior agles i polygos are just the iside agle at each vertex, ad they sum to differet amouts depedig o how may sides the polygo has. Aswers: This smaller agle may be acute, right-agled or obtuse but ever reflex ad Other times are probably ot exact. The hour had will have moved a bit as well. Less. Less: 1.20, 2.25, 3.30, 4.35, 5.40, 6.45, etc. More: 4.05, 5.10, 6.15, 7.20, 8.25, 9.30, etc. Hour had will be half way betwee 9 ad 10 (15 above the 9) so the agle is = 105. Aswers: Make use of symmetry (3.30 will be the same agle as 8.30 ot 9.30 because it s just a reflectio i a mirror). Be sure that pupils do t thik you re sayig some of the agles i a triagle are 180 ad some are t! You ca make these quickly ad easily usig two or three sheets of coloured A4 paper ad a guillotie. Make sure that there are a variety of acute-agled, obtuse-agled ad right-agled triagles. Tearig is better tha cuttig because it s easy to see which corer is the agle that was previously i the triagle.

4 Optio 2 may be better if may pupils have doe optio 1 before. Optio 2: Put the logest side horizotal (largest agle at the top) if you ca tell, ad measure the legths of the two shorter sides; divide those legths by two to fid the mid-poits of those two shorter sides ad mark them. Joi them with a ruler to give a lie parallel to the bottom side ad fold the top part dow alog this lie (see diagram right). Fold i the other two corers (vertical fold lies) ad all three agles should meet together ad make 180. Note that either optio 1 or optio 2 is a proof, although they ca be tured ito proofs by thikig about equal agles. Dyamic Proof: Draw a geeral triagle o the board ad put a large cardboard arrow or similar straight object (must have distiguishable eds) i the middle of the bottom side (see diagram right, 1). We re goig to use the arrow to add up agles. Slide it to the left corer (o chage i directio so o agle yet) ad rotate it to measure the iterior agle at that corer (2). Slide it to the top corer, ad, still rotatig aticlockwise, add o that corer agle (actually the agle vertically opposite to it, which is equal) (3). Slide it, without chagig directio, to the fial (right) corer ad add o that agle (still aticlockwise), leavig the arrow poitig 180 relative to its iitial positio (1) (This is similar to the stadard proof that the exterior agles of ay polygo add up to 360 by walkig aroud it.) How may times i the course of 12 hours are the hads of a clock at right agles to each other? Pupils ca ivet similar puzzles The hour had ad miute had o a aalogue clock coicide at 12 oo. Whe is the ext time whe they coicide exactly? Hit: It wo t be exactly 1.05 pm. Will it be before or after? (After) If they coicide t hours after 1 o clock, the 30( t 1) ad 360t, so solvig h m simultaeously gives 30( t 1) 360t ad so hour after 1 o clock. 1 t Triagles. Draw up a table like this (big eough to cotai drawigs): Work like this shows the dyamic aspect of agles they re a measure of turig movemet. Aswer: 22 times Imagie startig ad fiishig at The hads will be at right agles twice i every oe hour period, except that we will cout 3.00 ad 9.00 twice (because they occur o the hour), so we have to subtract those two = 22 times. Aswer: Ca use simultaeous equatios, eve the cocept of agular velocity, if you like. A eater way is to see first that the occasios whe the hads coicide will occur regularly. (At the ish time, imagie rotatig the paited umbers so that it reads 12 oo agai you could carry o like this.) Sice it will happe 11 times i 12 hours, each coicidece will occur after 12 of a hour; i.e., at ad 27 secs, 2.10 ad 55 secs, etc. There are two systems for amig triagles: by their agles or by the legths of their sides. acuteagled obtuse- scalee isosceles equilateral Note that acute-agled meas all the agles are acute, whereas obtuse-agled meas oly that there is oe obtuse agle (more would be impossible

5 agled rightagled For the top left square, if a triagle ca be both scalee ad acute-agled, draw a example. Put X if it s impossible, ad try to say why. Complete the table What sorts of agles ca a triagle have? Here are some more specific questios. If the aswer is yes, make up a example ad draw or list the agles; if the aswer is o, try to explai why ot. Ca a triagle have (as iterior agles) 1. a obtuse agle? 2. two obtuse agles? 3. a obtuse agle ad a right agle? 4. a obtuse agle ad a acute agle? 5. a reflex agle; 6. two right agles; 7. a right agle ad a acute agle? 8. four acute agles? I m thikig of a triagle. Tell me how big its agles are ad what kid of triagle it is. I ll call the agles i each oe A, B ad C. 1. Agle B is twice the size of agle A ad agle C is three times the size of agle A; 2. Agle A is twice the size of agle B ad agle C is the same size as agle B; 3. Agle B is twice the size of agle A ad agle C is three times the size of agle B (ot same as questio 1); 4. Agle B is four times the size of agle A ad agle C is five times the size of agle A. see sectio ). Aswers: Equilateral triagles must be acute-agled, because all the agles have to be 60. All other combiatios are possible. Aswers: 1. yes, lots of examples; 2. o, because their total would be >180 ; 3. o, because their total would be >180 ; 4. yes, lots of examples; 5. o, because it would be >180 ; 6. o, because their total would be 180 ad there eeds to be a o-zero third agle; 7. yes, lots of examples; 8. o, because although their total might be 180 (e.g., 40, 45, 45, 50 ), a tri-agle has to have exactly 3 agles! Note that we are always talkig about the iterior agles of the triagles. Aswers: 1. A = 30, B = 60, C = 90 ; scalee right-agled triagle; 2. A = 90, B = 45, C = 45 ; isosceles right-agled triagle; 3. A = 20, B = 40, C = 120 ; scalee obtuse-agled triagle; 4. A = 18, B = 72, C = 90 ; scalee right-agled triagle. These ca be solved by formig equatios or by trial ad improvemet or by ituitio. Pupils ca make up their ow for each other Polygo Agles. ( Iterior agles i polygos meas somethig differet from iterior agles formed whe a straight lie crosses a pair of parallel lies.) What do the agles iside a square add up to? What if I just draw ay quadrilateral? (Keep it covex for ow.) Drawig polygos with various umbers of sides ad measurig ad summig the iterior agles of each teds to give very iaccurate results, although it is a possible approach. Most will kow/guess 360. Show how it ca be split by a diagoal ito two triagles. Colour the agles i oe triagle red ad the other blue. What do the blue agles add up to?, etc. Use this trick of splittig ito triagles to work out the total iterior agle i polygos with sides from 5 all the way up to 10. Choose oe vertex ad joiig this to all the other vertices, so dividig the polygo ito triagles. If the polygo happes to be regular, what ca you say about each of the iterior agles? If the polygo cotais oe or more reflex agles, If the polygo has sides (ad so vertices), this method will divide it ito 2 triagles. So the total iterior agle = 180( 2). I a regular polygo, the iterior agles will be 180( 2) 360 equal, so each will be 180.

6 this method of dissectio ito triagles does t work. I that case, oe or more poits have to be chose iside the triagle ad these joied to as may vertices as possible. I this way, the polygo ca always be divided ito triagles, ad each additioal iteral poit cotributes a extra 360 to the total agle, ad this has to be subtracted. (The total iterior agle for a cocave polygo like this is always the same as for a covex polygo with the same umber of sides.) Plot a graph of size of oe iterior agle of a regular -go agaist. Make a predictio for 100. Why does this happe? o. of sides o. of triagles 2 total iterior agle size of each agle if polygo is regular The graph gets closer ad closer to 180 as the umber of sides the polygo has goes up. (There is a asymptote at 180.) Whe 100, agle = As the umber of sides goes up, the polygo looks more ad more like a circle, ad if you zoom i o ay part of the circumferece of a circle it looks almost like a straight lie (180 iterior agle). Mathematically, if is the agle, the 180( 2) , ad as gets larger ad larger, 360 gets smaller ad smaller, so that icreases, gettig closer ad closer to 180. (As, 0.) Exterior agles. NEED ewspaper ad scissors. The exterior agle does t mea all the agle outside the shape at each vertex; it meas the chage i directio at each vertex, so that exterior agle + iterior agle at each vertex equals 180 (see diagram right). (For a cocave polygo, where the iterior agle at oe or more vertices is a reflex agle, we say that there is o exterior agle at that vertex.) Place sheets of ewspaper o the floor ad imagie walkig roud it (start i the middle of oe of the sides). The total chage i directio is 4 90 = 360. This will always happe. (Cut the paper so that it makes a more uusual polygo ad do it agai.) After we ve walked roud the whole shape we re always poitig back i the same directio. Commo-sese says the agle at each corer will get bigger as the polygo gets more sides. A alterative way to fid the iterior agle of a regular polygo is to begi with the exterior agles (which sum to 360 for ay polygo), so i a regular polygo each must be 360. Therefore, sice each exterior agle ad iterior agle together must make a straight lie, each iterior agle must be equal to ( 2) 180. exterior agle (Or you ca go outdoors ad use or draw with chalk a shape o the groud.) For this to work with a cocave polygo we eed to cout the exterior agle where there is a reflex iterior agle as egative, because at that corer we chage from goig clockwise to goig aticlockwise (or vice versa). With a equilateral triagle, the exterior agles are 120. Exterior agles are importat whe doig LOGO programmig (see sectio 3.8) Ay covex polygo with sides ca be split up ito 2 triagles. How may ways are there of doig that? Aswer: o. of ways o. of ways

7 They fit the formula umber of ways = 2 4 C 2 1 For istace, for 6 you ca have, e.g., These are the Catala umbers (Eugèe Catala, ). It might seem that for 5 there will just be ways, because 3 vertices ca be joied to each of the vertices, but there are more tha this because the cuts eed ot all meet at oe poit. For 6 there are dissectios like those o the left Costructig ad solvig equatios from polygo ad parallel lie agles; e.g., a triagle has agles 2x, 3x ad 5x ; how much are they? Agles i 3 dimesios. Two diagoals are draw o two adjacet faces of a cube. If they meet at a vertex, what agle do they make with each other? Aswer: 10x 180, so x 18, so the three agles are 36, 54 ad 90. It s easy to make up this kid of thig or fid examples i books. Aswer: eed to do a diagram or make a model. People ofte go for complicated methods like trigoometry or vectors, but particularly i geometry there is ofte a easier ad more elegat solutio How much ca a object lea without topplig? For istace, a cuboid brick stadig o its ed what s the steepest slope it ca balace o? For this brick, 1 1 maximum agle of slope = ta If you add i a third diagoal, you ca see that you get a equilateral triagle (all sides are the same legth), so the iterior agles are all 60. Aswer: If the object is uiform (the same all the way through), the its cetre of mass will lie at the geometric cetre. The object will be stable if a vertical lie goig through the cetre of mass passes withi the base. (We assume that there is plety of frictio so the block wo t slide dow the slope.)

8 Which Agles are Equal? Use arrows to show which lies below are parallel to each other. Every time two lies cross each other, they create four agles. There are 15 crossig poits i the drawig below, so there are 60 agles. Use colour to show which agles are the same size as each other.