5 ANGLES AND POLYGONS

Size: px

Start display at page:

Download "5 ANGLES AND POLYGONS"

Rudolf Norton
6 years ago
Views:

1 5 GLES POLYGOS urling rige looks like onventionl rige when it is extene. However, it urls up to form n otgon to llow ots through. This Rolling rige is in Pington sin in Lonon, n urls up every Friy t miy. 60 Ojetives efore you strt In this hpter you will: reognise n use orresponing ngles n lternte ngles use n prove ngle properties of tringles n qurilterls reognise ngles of elevtion n epression give resons for ngle lultions reognise n know the nmes of speil polygons know n use the interior n exterior ngle properties of polygons fin the ering of one point from nother. You shoul know: how to mesure n rw ngles to the nerest egree how to fin the sizes of missing ngles on stright line n t the intersetion of stright lines how to reognise perpeniulr n prllel lines, n vertilly opposite ngles tht the ngle sum of tringle is 180 how to reognise slene, isoseles, equilterl n right-ngle tringles, n use their ngle properties how to reognise ute, otuse, reflex n right ngles tht qurilterl is shpe with four stright sies n four ngles.

5.1 ngle properties of prllel lines 5.1 ngle properties of prllel lines Ojetives You n mrk prllel lines on igrm. You n reognise orresponing n lternte ngles.

2 5.1 ngle properties of prllel lines 5.1 ngle properties of prllel lines Ojetives You n mrk prllel lines on igrm. You n reognise orresponing n lternte ngles. You n fin the sizes of missing ngles using orresponing n lternte ngles. You n give resons for ngle lultions. Get Rey 1. Fin the sizes of the mrke ngles f 84 Key Points When two prllel lines re rosse y stright line, s in the igrm, ngles re forme. The green ngles re in orresponing positions so they re lle orresponing ngles. orresponing ngles re equl. The lue ngles re lso orresponing ngles. The ornge ngles re on opposite or lternte sies of the line so they re lle lternte ngles. The yellow ngles re lso lternte ngles. The ornge ngles re on opposite or lternte sies of the line so they re lle lternte ngles. lternte ngles re equl. The yellow ngles re lso lternte ngles. Exmple 1 Fin the size of ngle n ngle so 67 orresponing ngles re equl. lternte ngles re equl. Prllel lines re mrke with rrows in igrms. prllel orresponing ngles lternte ngles 61

3 hpter 5 ngles n polygons Exmple 2 Explin why 180º. In the igrm 180 so 180 orresponing ngles re equl. ngles on stright line up to 180. Exerise 5 Questions in this hpter re trgete t the gres inite. In questions 1 6 fin the size of eh lettere ngle. Give resons for your nswers r m 63 q 67 p 81 l n 4 58 z 5 g 6 32 k y h 57 7 Here re two prllel lines rosse y stright line. List pirs of equl orresponing ngles. List pirs of equl lternte ngles. List pirs of ngles whih up to 180. Explin why the ngles up to 180. p s q r O3 8 E is stright line. Explin why the lines n must e prllel E 62

4 5.2 Proving the ngle properties of tringles n qurilterls 5.2 Proving the ngle properties of tringles n qurilterls Ojetives You n unerstn proof tht the ngle sum of tringle is 180. You n unerstn proof tht n exterior ngle of tringle is equl to the sum of the interior ngles t the other two verties. You n explin why the ngle sum of qurilterl is 360. Why o this? In mthemtis it is importnt to e le to prove tht results re lwys true. emonstrtion only shows tht the result is true for the hosen vlues. Get Rey In questions 1 3, lulte the size of eh lettere ngle. Give resons for your nswers e Key Points The ngle mrke e is lle n exterior ngle. The ngle of the tringle t this vertex, i, is sometimes lle n interior ngle. i e 180 i e Exmple 3 p Prove tht the ngle sum of ny tringle is 180. For ny tringle, stright line n e rwn through vertex prllel to the opposite sie, s shown in the igrm. p q lternte ngles re equl. 03 q lternte ngles re equl. p q 180 ngles on stright line up to 180. So 180 The ngle sum of tringle is 180. exterior ngle interior ngle 63

5 hpter 5 ngles n polygons Exerise 5 O3 1 Here is tringle with one sie extene. e omplete the following proof tht e y giving reson for eh line of the proof. 180 e 180 so e n exterior ngle of tringle is equl to the sum of the interior ngles t the other two verties. O3 2 Here is qurilterl. igonl hs een rwn to ivie the qurilterl into two tringles. p q r s opy n omplete this proof tht the ngle sum of qurilterl is 360. p q r s p r 180 q s 180 ing p q r s so 360 The sum of the ngles of qurilterl is 360. O3 3 Use properties of prllel lines to prove tht Whih ngle property of tringles hs this prove? 64

6 5.3 Using the ngle properties of tringles n qurilterls 5.3 Using the ngle properties of tringles n qurilterls Ojetives You n use the property tht n exterior ngle of tringle is equl to the sum of the interior ngles t the other two verties. You n use the property tht the ngle sum of qurilterl is 360º. You n use the ngle properties of prllelogrm. You n give resons for ngle lultions. Why o this? The ngles in tringles re use in sports, for exmple in wter-skiing. To ensure the longest jumps re me, the ngle of the jump shoul e 14º to the wter. Get Rey 1. lulte the size of the ngle mrke j. 106 j Key Points n exterior ngle of tringle is equl to the sum of the interior ngles t the other two verties. The ngle sum of qurilterl is º Opposite ngles of prllelogrm re equl. The two ngles t the en of eh sie of prllelogrm up to 180º. 180 Exmple 5 shows importnt ngle properties of prllelogrms. 65

hpter 5 ngles n polygons Exmple 4 is stright line. ngle 123 ngle 58 Work out the size of ngle. 123 58 123 ngle is n exterior ngle of the tringle.

7 hpter 5 ngles n polygons Exmple 4 is stright line. ngle 123 ngle 58 Work out the size of ngle ngle is n exterior ngle of the tringle. ngle Exterior ngle is equl to the sum of the interior ngles t the other two verties. 58 Exmple 5 is prllelogrm. Fin the size of eh ngle of the prllelogrm lternte ngles on prllel lines n re equl. ngles on stright line up to orresponing ngles on prllel lines n re equl. ngle sum of qurilterl is 360. Exerise 5 1 E is qurilterl. E is n equilterl tringle. Work out the size of ngle. E O2 O3 2 Work out the size of the ngle e. 76 Give resons for your working. e

8 5.4 ngles of elevtion n epression 3 Here is kite. The igonl shown otte is n xis of symmetry. Fin the size of ngle n the size of ngle. Give resons for your working. 132 O3 59 Here is n isoseles trpezium. The line shown otte is n xis of symmetry. The trpezium hs n ngle of 66 s shown. Fin the sizes of the three other ngles of the trpezium. Give resons for your working Here is qurilterl. Work out the size of ngle. Give resons for your working O ngles of elevtion n epression Ojetives You n reognise n ngle of elevtion. You n reognise n ngle of epression. You know tht the ngle of elevtion of point from point is equl to the ngle of epression of point from point. Why o this? If you wnte to seil own uiling, you oul work out the height of the uiling using the ngle of elevtion from point on the groun to the top. Get Rey 1. Explin why. Wht is? Key Points The ngle of elevtion of point from point is the ngle of turn ove the horizontl to look iretly from to. ngle of elevtion horizontl ngle of elevtion 67

9 hpter 5 ngles n polygons The ngle of epression of point from point is the ngle of turn elow the horizontl to look iretly from to. horizontl ngle of epression ngles of elevtion n epression re lwys mesure from the horizontl. horizontl Exerise 5 O3 1 Here is igrm of lighthouse. Explin why ngle e ngle. horizontl horizontl e 2 The ngle of epression of point on horizontl groun from the top of tree is 30. Show this informtion in sketh. On your sketh show n lel the ngle of elevtion of the top of the tree from. 5.5 erings Ojetives You n use erings for iretions. You n fin the ering of one point from nother point. You n work out the ering of point from point when you know the ering of point from point. Why o this? When giving iretions, it is importnt to know the ext iretion of from. This is useful for orienteering. Get Rey Yuen lives in town. The igrm shows the position of three ples in the town in reltion to Yuen s home. Give the ompss iretions from Yuen s home of: shool the inem the prk W E the shool. home inem W E SW S SE prk 68 ngle of epression

10 5.5 erings Key Points erings re ngles mesure lokwise from orth. erings lwys hve three figures. Exmple 6 For eh igrm, give the ering of from. Wth Out! Rememer tht ering hs to hve three figures n tht is why some strt with one or two zeros. t, turn 47 lokwise from orth to look towrs. The ering of from is 047. The ngle 351 is mesure ntilokwise from orth. lokwise ngle (s there re 360 in omplete turn). The ering of from is Exmple 7 The ering of from is 107. Work out the ering of from. Exminer s Tip ering of from º Rememer tht erings re lwys mesure lokwise from the orth rw orth line t. The ngle mrke in re is the ngle tht is the ering of from. The ngle mrke in lue is 106 (lternte ngles). erings 69

11 hpter 5 ngles n polygons Exerise 5E 1 In eh of the following, give the ering of from The igrm shows three towns, n The ering of from is 038. ngle 116 Work out the ering of from from from 3 The ering of orwih from Glouester is 069. Work out the ering of Glouester from orwih. O2 4 plne flies from Skegness to rlisle on ering of 132. Work out the ering the plne nees to fly on for the return journey to Skegness. O2 O3 5 The igrm shows the position of three towns P, Q n R. Fin the ering of: R from Q P from Q. 121 Q 111 P R 70

12 5.6 Using ngle properties to solve prolems 5.6 Using ngle properties to solve prolems Ojetives You n use the ngle properties in this hpter to solve more involve prolems. You n give resons for ngle lultions. Why o this? In the fshion inustry, you n use ngle properties to fit piees of mteril together for lothes without wsting loth. Get Rey 1. Fin the size of eh lettere ngle. Give resons for your nswers g 43 e 116 f Exmple 8 Work out the size of: i ngle ii ngle. i ii Sum of the ngles of qurilterl is 360. lternte ngles re equl. 103 Wth Out! Questions will rrely just sk you to work out the size of the unknown ngle of tringle or of qurilterl. In most ses you will nee to use other ngle properties. Exerise 5F 1 F n IG re prllel lines. EFGH is stright line. GH GI ngle GIH 28 Work out the size of: ngle GF ngle EF. E F I G 28 H O2 O3 71

13 hpter 5 ngles n polygons 2 L, M n re points, s shown, on the sies of tringle. ML n re prllel. L n re prllel. M n re prllel. ngle 70 ngle 55 Work out the size of eh ngle of tringle LM. 70 M L O2 O3 3 Work out the size of: ngle p ngle q. Give resons for your working. 58 p q O2 O3 4 Here is prllelogrm. Explin why. Hene prove tht. Give resons for your working. Wht property of prllelogrms hve you prove? O2 O3 5 Here is qurilterl. In this qurilterl 180. Prove tht 180. Give resons for your working. 5.7 Polygons Ojetives You know wht polygon is n wht regulr polygon is. You know the nmes of speil polygons. You know n n use the sum of the interior ngles of polygon. You know n n use the sum of the exterior ngles of polygon. You n nswer prolems on polygons involving ngles. Why o this? Tiles re me in the shpe of regulr polygons. Knowing the properties of vrious polygons might help you mke ptterns with ifferent shpes. Get Rey 1. Write own the nme of the shpe tht hs three equl sies n three equl ngles. 2. Use two wors to omplete the following sentene: squre is qurilterl with sies n ngles. 72

14 5.7 Polygons Key Points polygon is lose two-imensionl shpe with stright sies. regulr polygon is polygon with ll its sies the sme length n ll its ngles equl in size. Here re the nmes of some speil polygons. Tringle Qurilterl Pentgon 3-sie polygon 4-sie polygon 5-sie polygon Exminer s Tip These re the nmes of the polygons tht re neee for GSE. Other polygons lso hve speil nmes. Hexgon 6-sie polygon Otgon 8-sie polygon Heptgon 7-sie polygon egon 10-sie polygon ngles of polygon Exmple 9 Here is 7-sie polygon. Work out the sum of the ngles of this polygon. 7-sie polygon is heptgon. ll the igonls from one vertex (orner) of the heptgon hve een rwn. There re 4 igonls n the heptgon hs een ivie into 5 tringles. The ngle sum of eh tringle is 180. Sum of the ngles of heptgon s these ngles re insie the polygon, they re lso lle the interior ngles of the polygon. polygon regulr 73

15 hpter 5 ngles n polygons Exerise 5G 1 opy n omplete the following tle. Polygon umer of sies (n) umer of igonls from one vertex umer of tringles forme Sum of interior ngles Tringle º Qurilterl º Pentgon 5 Hexgon 6 Heptgon º Otgon 8 ongon 9 egon 10 For polygon with n sies, write own: i the numer of igonls tht n e rwn from one vertex ii the numer of tringles tht re forme iii the sum of the interior ngles of the polygon. O3 2 rhomus hs sies tht re the sme length. Explin why rhomus is, in generl, not regulr polygon. Sum of the interior ngles of polygon Key Points polygon n e ivie into tringles when ll igonls re rwn from one vertex. For n n-sie polygon, the numer of tringles will e (n-2). regulr polygon will tesselte if the interior ngle is n ext ivisor of 360. Regulr tringles, squres n hexgons will therefore tessellte. Sum of the interior ngles of polygon with n sies (n 2) 180 (2n 4) right ngles Exmple 10 polygon hs 15 sies. Work out the sum of the interior ngles of the polygon. Fin the size of eh interior ngle of regulr polygon with 15 sies With n 15, work out the numer of tringles (n 2) Sum of interior ngles 2340 Eh interior ngle Work out (n 2) 180 The regulr polygon hs 15 interior ngles tht re ll the sme size. 156 ivie the sum of the ngles y

16 5.7 Polygons Exmple 11 Here is regulr otgon with entre O. Work out the size of: i ngle x ii ngle y. Hene work out the size of eh interior ngle of regulr otgon. x O i x ii y Joining eh vertex of the polygon to the entre O will form in totl 8 equl ngles like ngle x. These 8 ngles mke omplete turn of 360. The tringle shown is n isoseles tringle, with ngle y s one of the two equl se ngles. y Eh interior ngle y symmetry, ngle y is hlf n interior ngle of the regulr polygon. Exerise 5H 1 John ivies regulr polygon into 16 tringles y rwing ll the igonls from one vertex. How mny igonls oes John rw? How mny sies hs the polygon? Wht is the size of eh of the interior ngles of the polygon? 2 Work out the size of eh interior ngle of: regulr hexgon regulr egon regulr polygon with 30 sies. 3 Work out the size of eh of the mrke ngles in these polygons. You must show your working y O2 O3 O2 O3 O x 4 Explin why the size of the ngle t the entre of regulr polygon nnot e 25º. 5 Here is n otgon. Work out the size of eh of the ngles mrke with letter. Work out the vlue of e f g h h O2 O3 g f 136 e 75

hpter 5 ngles n polygons O2 O3 6 is squre. EFGHIJKL is n otgon. E EF F G GH H I IJ J K KL L Fin the size of eh interior ngle of the otgon. Give resons for your nswers.

17 hpter 5 ngles n polygons O2 O3 6 is squre. EFGHIJKL is n otgon. E EF F G GH H I IJ J K KL L Fin the size of eh interior ngle of the otgon. Give resons for your nswers. Try sys tht the otgon is regulr otgon. i Why might Try think tht the otgon is regulr? ii Explin why Try is wrong. K L J I E F H G Exterior ngles of polygon Key Points When sie of polygon is extene t vertex, the ngle etween this extene line n the other sie t the vertex is n exterior ngle t this vertex. interior ngle exterior ngle The sum of ngles on stright line 180 So t vertex, interior ngle exterior ngle 180 The sum of the exterior ngles of ny polygon is 360. EF is hexgon. Imgine spier is t vertex fing in the iretion of the rrow. The spier turns through ngle so tht it is now fing in the iretion. The spier now wlks to vertex. t, the spier turns through ngle to fe in the iretion. He ontinues to wlk roun the hexgon until he gets k to. The spier hs turne through one omplete irle, so it hs turne through n ngle of 360. The totl ngle turne through y the spier is lso e f, the sum of the exterior ngles of the hexgon. So e f 360. The sme rgument hols for ny polygon. f F e E Exmple 12 regulr polygon hs 20 sies. Work out the size of eh exterior ngle. Work out the size of eh interior ngle. Eh exterior ngle The polygon is regulr so the 20 exterior ngles re equl in size. The sum of these 20 equl ngles is 360. Interior ngle Exterior ngle interior ngle

18 5.7 Polygons Exmple 13 The interior ngle of regulr polygon is 160º. Work out how mny sies the polygon hs. Exterior ngle Work out the size of n exterior ngle. Exterior ngle interior ngle 180. umer of sies The polygon is regulr so tht ll exterior ngles re 20 with sum 360. Exerise 5I 1 One vertex of polygon is the point P. Work out the size of the interior ngle t P when the exterior ngle t P is: i 70 ii 37. Work out the size of the exterior ngle t P when the interior ngle t P is: i 130 ii Work out the size of eh exterior ngle of: regulr pentgon regulr otgon regulr polygon with 12 sies regulr 25-sie polygon. 3 The size of eh exterior ngle of regulr polygon is 15. Work out the numer of sies the polygon hs. Wht is the sum of the interior ngles of the polygon? O2 O3 4 The sizes of five of the exterior ngles of hexgon re 36, 82, 51, 52 n 73. Work out the size of eh of the interior ngles of the hexgon. O3 5, n re three verties of regulr polygon with 30 sies. Work out the size of ngle. Give resons for your working. O2 O3 * 6 The igrm shows three sies,, n, of regulr polygon with entre O. The ngle t the entre of the polygon is. O The exterior ngle of the polygon t the vertex is e. Explin why e. O2 O3 e 77

19 hpter 5 ngles n polygons hpter review When two prllel lines re rosse y stright line, s in the igrm, ngles re forme. orresponing ngles re equl, n lternte ngles re equl. The ngle sum of tringle is 180. n exterior ngle of tringle is equl to the sum of the interior ngles t the other two verties. The ngle sum of qurilterl is º Opposite ngles of prllelogrm re equl. The ngle of elevtion of point from point is the ngle of turn ove the horizontl to look iretly from to. ngle of elevtion horizontl The ngle of epression of point from point is the ngle of turn elow the horizontl to look iretly from to. horizontl ngle of epression ngles of elevtion n epression re lwys mesure from the horizontl. erings re ngles mesure lokwise from orth. erings lwys hve three figures. polygon is lose two-imensionl shpe with stright sies. horizontl regulr polygon is polygon with ll its sies the sme length n ll its ngles equl in size. Sum of the interior ngles of polygon with n sies (n 2) 180 (2n 4) right ngles t vertex of polygon, interior ngle exterior ngle 180 The sum of the exterior ngles of ny polygon is

20 hpter review Review exerise 1 igrm OT urtely rwn Exm Question Report % of stuents nswere this question poorly euse they i not justify their nswers. Jmes sys, The lines n re prllel. en sys, The lines n re not prllel. Who is right, Jmes or en? Give reson for your nswer. My is stright line. This igrm is wrong. Explin why igrm OT urtely rwn ov Write own the vlue of x. Give reson for your nswer. x 30 igrm OT urtely rwn This igrm is wrong. Explin why igrm OT urtely rwn June is n isoseles tringle. igrm OT is stright line. urtely rwn. 54 ngle 54. i Work out the size of the ngle ii mrke x. x y Give reson for your nswer. Work out the size of the ngle mrke y. June The igrm shows the position of two ots, P n Q. The ering of ot R from ot P is 060. The ering of ot R from ot Q is 310. rw n urte igrm to show the position of ot R. Mrk the position of ot R with ross ( ). Lel it R. Q P June

21 hpter 5 ngles n polygons 6 is prllel to. Write own the vlue of y. Give reson for your nswer y June 2008 * 7 P R y 50 x is stright line. PQ is prllel to RS. Write own the size of ngle x n y, Q S Exm Question Report 92% of stuents nswere this question well euse they knew the ifferene etween lternte n orresponing ngles. giving resons for your nswer. Mrh Work out the vlue of x x ov In the igrm x ngle 52 Work out the size of the ngle mrke x. 52 Give reson for your nswer. June EF is regulr hexgon n QP is squre. ngle Q x. Work out the vlue of x. F E x P Q June

22 hpter review 11 In tringle, ngle 90. Work out the size of ngle ngle. O2 O3 12 Work out the size of the ngle p. Give resons for your working. 127 O2 O3 p S w y igrm OT urtely rwn 25 x P Q R PQR is stright line. PQ QS QR. ngle SPQ 25. i Write own the size of ngle w. ii Work out the size of ngle x. Work out the size of ngle y. Exm Question Report 84% of stuents nswere this question poorly euse they i not use ll of the informtion given in the question. ov Fin the ering of from. On opy of the igrm, rw line on ering of 135 from. ov The igrm shows prt of regulr 10-sie polygon. Work out the size of the ngle mrke x. x O2 ov

23 hpter 5 ngles n polygons 16 In the igrm, is stright line n. E Work out the size of ngle x. 108 Work out the size of ngle y y x ov In tringle,. ngle (x 20 ). Show tht ngle (140 2x). Give resons for eh stge of your working. igrm OT urtely rwn 18 Prove tht ngle FE 205 3x Give resons for eh stge of your working. E y 65 x 2x F igrm OT urtely rwn 19 In regulr polygon eh exterior ngle is two thirs the size of eh interior ngle. lulte the size of eh interior ngle. lulte the numer of sies of the polygon. * 20 In the igrm,. Prove tht ngle 4(95 x) Give resons for eh stge of your working. igrm OT urtely rwn x x 80 * 21 PQR is tringle with PQ PR. Prove tht PY QY. X Z Q igrm OT urtely rwn P 20 Y R O2 O3 22 Just fter 1 o lok the hour n minute hns of lok re pointing in the sme iretion, mening tht the ngle etween them is 0. Wht time is this? (nswer to the nerest seon). 82

10.2 Graph Terminology and Special Types of Graphs

10.2 Graph Terminology and Special Types of Graphs 10.2 Grph Terminology n Speil Types of Grphs Definition 1. Two verties u n v in n unirete grph G re lle jent (or neighors) in G iff u n v re enpoints of n ege e of G. Suh n ege e is lle inient with the