# State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127

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1 ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not proof. Tht omes lter in this hpter. The ngles on stright line dd up to 180 nd so the ngles in this tringle dd up to 180. The ngle sum of tringle is 180. Emple 1 Work out the size of ngle. Solution dd 72 nd Tke the result wy from 180, s the ngle sum of tringle is 180. Stte the size of ngle. Sometimes the ft tht the ngle sum of tringle is 180 nd other ngle fts re needed. Emple 2 Work out the size of ngle ngle y. Give resons for your nswers. 60 Solution y y ngle sum of tringle is 180. Sum of ngles on stright line is

2 CHTER 2 ngles Eerise 2 In this eerise, the tringles re not urtely drwn. In Questions 1 12,find the size of eh of the ngles mrked with letters nd show your working d e j 36 h i 43 g f 76 l k m n p q s r 123 v 114 t u 41 In Questions 13 15,find the size of eh of the ngles mrked with letters nd show your working. Give resons for your nswers w y 24 z Equilterl tringles nd isoseles tringles n equilterl tringle hs three equl sides nd three equl ngles. s the ngle sum of tringle is 180, the size of eh ngle is n isoseles tringle hs two equl sides nd the ngles opposite the equl sides re equl. tringle whose sides re ll different lengths is lled slene tringle

3 2.2 Equilterl tringles nd isoseles tringles CHTER 2 Emple 3 Work out the size of ngle ngle y. Give resons for your nswers. y 41 Solution 3 41 Isoseles tringle with equl ngles opposite equl sides ngle sum of tringle is 180. y 98 Emple 4 Work out the size of ngle. Give resons for your nswer. Solution ngle sum of tringle is Isoseles tringle with equl ngles opposite equl sides. 17 Eerise 2B In this eerise, the tringles re not urtely drwn. In Questions 1 12,find the size of eh of the ngles mrked with letters nd show your working d 29 e h i g f m l k j q u v p n 106 s r t

4 CHTER 2 ngles In Questions 13 15,find the size of eh of the ngles mrked with letters nd show your working. Give resons for your nswers w 42 y 2.3 Qudrilterls qudrilterl is shpe with four stright sides nd four ngles. Ter off its orners. Fit ngles,, nd d together t point. To find the ngle sum of qudrilterl, drw qudrilterl on pper nd lel its ngles,, nd d. d d d The ngles t point dd up to 360 nd so this shows tht the ngles in this qudrilterl dd up to 360. The ngle sum of qudrilterl is 360. To prove this result, drw digonl of the qudrilterl. The digonl splits the qudrilterl into two tringles. The ngle sum of eh tringle is 180. So the ngle sum of the qudrilterl is Emple 5 Work out the size of ngle Solution dd 76, 118 nd Tke the result wy from 360, s the ngle sum of qudrilterl is Stte the size of ngle. 20

5 2.3 Qudrilterls CHTER 2 Emple 6 Write down the size of ngle. Work out the size of ngle y. Give reson for eh nswer. Solution Where two stright lines ross, the opposite ngles re equl. y ngle sum of qudrilterl is 360 y 92 Emple 7 The digrm shows kite. Write down the size of ngle. Work out the size of ngle y. Solution kite hs line of symmetry y Give reson for eh nswer. ngle is refletion of the 109 ngle nd so the two ngles re equl y y 59 ngle sum of qudrilterl is 360. Eerise 2C In this eerise, the qudrilterls re not urtely drwn. In Questions 1 12,find the size of eh of the ngles mrked with letters nd show your working d e g 48 f h i 147 j n r 82 k 116 l 94 m p q 84 u s t

6 CHTER 2 ngles 13 The digrm shows kite. 14 The digrm shows kite. Write down the size of ngle v. Work out the vlue of. Work out the size of ngle w. 126 w v 15 The digrm shows n isoseles trpezium. Write down the vlue of. Work out the vlue of. In Questions 16 20, find the sizes of the ngles mrked with letters nd show your working. Give resons for your nswers h 74 e g f 61 k 113 i j m n l 2.4 olygons polygon is shpe with three or more stright sides. Some polygons hve speil nmes. 3-sided polygon is lled tringle. 4-sided polygon is lled qudrilterl. 5-sided polygon is lled pentgon. 6-sided polygon is lled hegon. n 8-sided polygon is lled n otgon. 10-sided polygon is lled degon. To find the sum of the ngles of polygon, split it into tringles. For emple, for this hegon, drw s mny digonls s possile from one orner. This splits the hegon into four tringles. The ngle sum of tringle is 180 nd so the sum of the ngles of hegon is Sometimes, these ngles re lled interior ngles to emphsise tht they re inside the polygon. 22

7 2.4 olygons CHTER 2 Using this method, the sum of the interior ngles of ny polygon n e found. umer of sides umer of tringles Sum of the interior ngles The numer of tringles into whih the polygon n e split up is lwys two less thn the numer of sides. Emple 8 Find the sum of the ngles of 12-sided polygon (dodegon). Solution Sutrt 2 from the numer of sides to find the numer of tringles The sum of the ngles 1800 Multiply the numer of tringles y 180. Stte the sum of the ngles in degrees. polygon with ll its sides the sme length nd ll its ngles the sme size is lled regulr polygon. So squre is regulr polygon, euse ll its sides re the sme length nd ll its ngles re 90, ut rhomus is not regulr polygon. lthough its sides re ll the sme length, its ngles re not ll the sme size. Here re three more regulr polygons. regulr pentgon regulr hegon regulr otgon The entgon in Wshington DC is the hedqurters of the US Deprtment of Defene. Bees honeyom is mde up of regulr hegons. Regulr otgons tessellte with squres. 23

8 CHTER 2 ngles Emple 9 Find the size of eh interior ngle of regulr degon. Solution Eh interior ngle is 144 Sutrt 2 from the numer of sides to find the numer of tringles. Multiply the numer of tringles y 180 to find the sum of ll 10 interior ngles. ll 10 interior ngles re the sme size. So divide 1440 y 10 Stte the size of eh interior ngle. Emple 10 The digrm shows regulr 9-sided polygon (nongon) with entre O. Work out the size of i ngle ii ngle y. Use your nswer to prt ii to work out the size of eh interior ngle of the polygon. y O Solution 10 i (40 is the ngle t the entre of regulr 9-sided polygon.) ii y 70 Eh interior ngle is 140. Eh orner of the polygon ould e joined to the entre O to mke 9 equl ngles t O. The totl of ll 9 ngles is 360, s ltogether they mke omplete turn. Stte the size of ngle. The ngle sum of tringle is 180 nd so the sum of the two se ngles is 140. The tringle is isoseles nd so the two se ngles re equl. Stte the size of ngle y. Beuse the polygon is regulr, it hs nine lines of symmetry nd eh interior ngle is twie the size of eh se ngle of the tringle. Stte the size of eh interior ngle. Eerise 2D In this eerise, the polygons re not urtely drwn. 1 Find the sum of the ngles of 15-sided polygon. 2 Find the sum of the ngles of 20-sided polygon. 3 polygon n e split into 17 tringles y drwing digonls from one orner. How mny sides hs the polygon? 24

9 2.4 olygons CHTER 2 In Questions 4 9,find the size of eh of the ngles mrked with letters nd show your working d e f The digrm shows pentgon. ll its sides re the sme length. Work out the vlue of g. Is the pentgon regulr polygon? Eplin your nswer. 60 g g 11 Work out the size of eh interior ngle of regulr pentgon regulr hegon regulr otgon. 12 Work out the size of eh interior ngle of regulr 15-sided polygon. 13 Work out the size of eh interior ngle of regulr 20-sided polygon. 14 Work out the size of the ngle t the entre of regulr pentgon. 15 Work out the size of the ngle t the entre of regulr 12-sided polygon. ustrli s 50 ent oin is regulr 12-sided polygon (dodegon) 16 The ngle t the entre of regulr polygon is 60. How mny sides hs the polygon? 17 The ngle t the entre of regulr polygon is 20. How mny sides hs the polygon? Work out the size of eh interior ngle of the polygon. 18 Work out the ngle t the entre of regulr otgon. Drw irle with rdius of 5 m nd, using your nswer to prt,drw regulr otgon inside the irle. 25

10 CHTER 2 ngles 19 Work out the ngle t the entre of regulr 10-sided polygon. Drw irle with rdius of 5 m nd, using your nswer to prt,drw regulr 10-sided polygon inside the irle. 20 The digrm shows pentgon. 21 The digrm shows hegon. Work out the size of Work out the size of ngle h ngle i. ngle j ngle k h i 68 j k 22 Crig sys, The sum of the interior ngles of this polygon is Eplin why he must e wrong. 23 The digrm shows qudrilterl. Work out the size of eh of the ngles mrked with letters. Work out l m n p n p m l 24 The digrm shows pentgon. Work out the size of eh of the ngles mrked with letters. t s r 81 Work out q r s t u 73 u 124 q 25 The digrm shows hegon. Work out the size of eh of the ngles mrked with letters. Work out u v w y z y 85 z u w v 2.5 Eterior ngles polygon s interior ngles re the ngles inside the polygon. Etend side to mke n eterior ngle,whih is outside the polygon. t eh verte (orner), the interior ngle nd the eterior ngle re on stright line nd so their sum is 180. interior ngle eterior ngle 180 The sum of the eterior ngles of ny polygon is 360. interior ngle eterior ngle 26

11 2.5 Eterior ngles CHTER 2 To show this, imgine someone stnding t on this qudrilterl, fing in the diretion of the rrow. Q They turn through ngle, so tht they re fing in the diretion Q, nd then wlk to Q. t Q, they turn through ngle, so tht they re fing in S d the diretion QR, nd then wlk to R. t R, they turn through ngle, so tht they re fing in the diretion RS, nd then wlk to S. t S, they turn through ngle d. They re now fing in the diretion of the rrow gin nd so they hve turned through 360. The totl ngle they hve turned through is lso the sum of the eterior ngles of the qudrilterl. So d 360 The sme rgument n e used with ny polygon, not just qudrilterl. R Emple 11 The sizes of four of the eterior ngles of pentgon re 67, 114, 58 nd 73. Work out the size of the other eterior ngle. Solution Eterior ngle 48 dd the four given eterior ngles. Sutrt the result from 360 Stte the size of the eterior ngle. Emple 12 For regulr 18-sided polygon, work out the size of eh eterior ngle, the size of eh interior ngle. Solution Beuse the polygon is regulr, ll 18 eterior ngles re equl. Their sum is 360 nd so divide 360 y 18 t orner, the sum of the interior ngle nd the eterior ngle is 180. So sutrt 20 from 180. Emple 13 The size of eh interior ngle of regulr polygon is 150. Work out the size of eh eterior ngle, the numer of sides the polygon hs. Solution t orner, the sum of the interior ngle nd the eterior ngle is 180. So sutrt 150 from 180. Beuse the polygon is regulr, ll the eterior ngles re 30. Their sum is 360 nd so divide 360 y 30 27

12 CHTER 2 ngles Eerise 2E 1 t verte (orner) of polygon, the size of the interior ngle is 134. Work out the size of the eterior ngle. 2 t verte of polygon, the size of the eterior ngle is 67. Work out the size of the interior ngle. 3 The sizes of three of the eterior ngles of qudrilterl re 72, 119 nd 107. Work out the size of the other eterior ngle. 4 The sizes of five of the eterior ngles of hegon re 43, 109, 58, 74 nd 49. Work out the size of the other eterior ngle. 5 Work out the size of eh eterior ngle of regulr otgon. 6 Work out the size of eh eterior ngle of regulr 9-sided polygon. 7 For regulr 24-sided polygon, work out the size of eh eterior ngle, the size of eh interior ngle. 8 For regulr 40-sided polygon, work out the size of eh eterior ngle, the size of eh interior ngle. 9 The size of eh interior ngle of regulr polygon is 168. Work out the size of eh eterior ngle, the numer of sides the polygon hs. 10 The size of eh interior ngle of regulr polygon is 170. Work out the numer of sides the polygon hs. 2.6 Corresponding ngles nd lternte ngles rllel lines re lwys the sme distne prt. They never meet. In digrms, rrows re used to show tht lines re prllel. In the digrm, stright line rosses two prllel lines. The shded ngles re lled orresponding ngles nd re equl to eh other. The F shpe formed y orresponding ngles n e helpful in reognising them. Other pirs of orresponding ngles hve een shded in the digrms elow. 28

13 2.6 Corresponding ngles nd lternte ngles CHTER 2 In the digrm, stright line rosses two prllel lines. The shded ngles re lled lternte ngles nd re equl to eh other. The Z shpe formed y lternte ngles n e helpful in reognising them. nother pir of lternte ngles hs een shded in this digrm. Emple 14 Write down the letter of the ngle whih is orresponding to the shded ngle, lternte to the shded ngle. Solution 14 ngle q is the orresponding ngle to the shded ngle. ngle s is the lternte ngle to the shded ngle. s r p q otie tht they form n F shpe. otie tht they form Z shpe. Emple 15 Find the size of ngle. Give reson for your nswer. 78 Solution lternte ngles. Emple 16 Find the size of ngle p. Give reson for your nswer. Find the size of ngle q. d Give reson for your nswer. 67 p q Solution p 113 The sum of the ngles on stright line is 180. q 113 d Corresponding ngles. 29

14 CHTER 2 ngles Eerise 2F In this eerise, the digrms re not urtely drwn. 1 Write down the letter of the ngle whih is orresponding to the shded ngle, lternte to the shded ngle. u w v 2 Write down the letter of the ngle whih is orresponding to the shded ngle, lternte to the shded ngle. y w v In Questions 3 5,find the sizes of the ngles mrked with letters nd stte whether the pirs of ngles re orresponding or lternte In Questions 6 20,find the sizes of the ngles mrked with letters. Give resons for your nswers e 132 d g f 82 i h k l m j n p q r t s 76 v w 47 u z 82 y d g e f 62 i h j 30

15 2.7 roofs CHTER k l m roofs In mthemtis, proof is resoned rgument to show tht sttement is lwys true. The proofs whih follow mke use of orresponding nd lternte ngles. roof 1 n eterior ngle of tringle is equl to the sum of the interior ngles t the other two verties The digrm shows tringle QR. Etend the side Q to S. t Q drw line QT prllel to R. Then ngle ngle (orresponding ngles) nd ngle y ngle (lternte ngles) dding, y y is the eterior ngle of the tringle nd is the sum of the interior ngles t the other two verties nd so the sttement is true. roof 2 The ngle sum of tringle is 180 This proof strts in the sme wy s roof 1. The digrm shows tringle QR. Etend the side Q to S. t Q drw line QT prllel to R. Then ngle ngle (orresponding ngles) nd ngle y ngle (lternte ngles) dding, y s, y nd re ngles on stright line, their ngle sum is 180, tht is So y whih proves tht the sttement is true. R R T y Q S T y Q S roof 3 The opposite ngles of prllelogrm re equl Drw digonl of the prllelogrm. ngle ngle (lternte ngles) ngle ngle d (lternte ngles) dding, d whih proves tht the sttement is true. d 31

16 CHTER 2 ngles Emple 17 Find the size of ngle w. Solution w 107 Give reson for your nswer. 63 w 44 Eterior ngle of tringle. (s the full reson is long, it my e shortened to this.) Emple 18 Find the size of ngle. Give reson for your nswer. Find the size of ngle y. d Give resons for your nswer. 71 y Solution Opposite ngles of prllelogrm re equl d ngle sum of qudrilterl is Opposite ngles of prllelogrm re equl y 109 Eerise 2G In this eerise, the digrms re not urtely drwn. Find the size of eh of the ngles mrked with letters. Give resons for your nswers d e h g i j k m n 64 f l p q r Berings Berings re used to desrie diretions. Berings re mesured lokwise from orth. When the ngle is less thn 100, one or two zeros re written in front of the ngle, so tht the ering still hs three figures. 32

17 2.8 Berings CHTER 2 Emple 19 Emple 20 Mesure the ering of B from. Mesure the ering of Q from. B Q Solution 19 From orth, mesure the ngle lokwise. The ngle is 52. So the ering is 052. Solution 20 To find the ngle lokwise from orth with semi-irulr protrtor mesure the shded ntilokwise ngle (38 ) nd sutrt it from B The ering of Q from is 322. Q Q Emple 21 Folkestone nd Dover re shown on the mp. The ering of ship from Folkestone is 117. The ering of the ship from Dover is 209. Drw n urte digrm to show the position of the ship. Mrk the position with ross X. Lel it S. Solution 21 Drw line on ering of 117 from Folkestone. Drw line on ering of 209 from Dover y mesuring n ngle of 151 ( ) ntilokwise from orth. (lterntively, mesure n ngle of 29 lokwise from South.) ut X where the lines ross. Lel the position S. Folkstone Folkstone 117 Dover Dover 209 S 33

18 CHTER 2 ngles Sometimes, nswers to questions hve to e worked out, not found using protrtor. Emple 22 The ering of B from is 061. Work out the ering of from B. B Digrm OT urtely drwn Solution 22 The ering of from B is the refle ngle t B. 61 y 61 (lternte ngles) Bering B y Digrm OT urtely drwn Eerise 11H In Questions 1 4, mesure the ering of Q from. 1 2 Q Q 3 4 Q Q 34

19 2.8 Berings CHTER 2 5 Drw digrms similr to those in Questions 1 4 to show the erings d The digrm shows two points, nd B. The ering of point L from is 048. The ering of L from B is 292. On the digrm on the resoure sheet find the position of L y mking n urte drwing. 7 The digrm shows two points, nd Q. The ering of point M from is 114. The ering of M from Q is 213. On the digrm on the resoure sheet find the position of M y mking n urte drwing. B Q 8 Cromer nd Gret Yrmouth re shown on the mp. The ering of ship from Cromer is 052. The ering of the ship from Gret Yrmouth is 348. On the digrm on the resoure sheet find the position of the ship y mking n urte drwing. Mrk the position of the ship with ross X. Lel it S. Cromer Gret Yrmouth 9 The ering of Q from is 038. Work out the ering of from Q. 10 The ering of T from S is 146. Work out the ering of S from T. Digrm OT urtely drwn Q S Digrm OT urtely drwn T 11 The ering of B from is 074. The ering of C from B is 180. B C. Work out the ering of C from. 74 B Digrm OT urtely drwn C 12 The digrm shows the positions of York, Srorough nd Hull. The ering of Srorough from York is 052. The ering of Hull from York is 118. The distne etween York nd Srorough is the sme s the distne etween York nd Hull. Work out the ering of Hull from Srorough. York 52 Srorough Digrm OT urtely drwn Hull 35

20 CHTER 2 ngles Chpter summry You should know nd e le to use these fts The ngle sum of tringle is 180. n equilterl tringle hs three equl ngles nd three equl sides n isoseles tringle hs two equl sides nd the ngles opposite the equl sides re equl. tringle whose sides re ll different lengths is lled slene tringle. qudrilterl is shpe with four stright sides nd four ngles. The ngle sum of qudrilterl is 360. polygon is shpe with three or more stright sides. 5-sided polygon is lled pentgon. 6-sided polygon is lled hegon. n 8-sided polygon is lled n otgon. 10-sided polygon is lled degon. The ngle sum of polygon n e found y sutrting 2 from the numer of sides nd multiplying the result y 180. polygon with ll its sides the sme length nd ll its ngles the sme size is lled regulr polygon. t verte, interior ngle eterior ngle 180. The sum of the eterior ngles of ny polygon is 360. To find the size of eh eterior ngle of regulr polygon, divide 360 y the numer of sides. interior ngle eterior ngle Where stright line rosses two prllel lines, the orresponding ngles re equl. Where stright line rosses two prllel lines, the lternte ngles re equl. Berings re mesured lokwise from orth. You should lso know these proofs n eterior ngle of tringle is equl to the sum of the interior ngles t the other two verties. The ngle sum of tringle is

21 Chpter 2 review questions CHTER 2 Chpter 2 review questions In Questions 1 12,find the size of eh of the ngles mrked with letter. The digrms re not urtely drwn d 56 f e h g i 112 k 94 j 141 l n m q p r s t In tringle BC, B C nd ngle C 50. Write down the speil nme of tringle BC. Work out the vlue of y. B y 50 Digrm OT urtely drwn C (1385 June 1999) 14 Clulte the vlue of Digrm OT urtely drwn (4400 ovemer 2004) 15 Work out the vlue of Digrm OT urtely drwn (1388 Mrh 2002) 16 Work out the size of eh eterior ngle of regulr 10-sided polygon. 17 Work out the sum of the interior ngles of 9-sided polygon. The size of eh eterior ngle of regulr polygon is 20. Work out how mny sides the polygon hs. 37

22 CHTER 2 ngles 18 The digrm shows regulr hegon. Work out the vlue of. Work out the vlue of y. y Digrm OT urtely drwn (1385 June 2001) 19 BCDE is regulr pentgon. EF nd CDF re stright lines. Work out the size of ngle DFE. Give resons for your nswer. E B Digrm OT urtely drwn F D C (1388 Mrh 2004) 20 i Write down the size of the ngle mrked. ii Give reson for your nswer. i Write down the size of the ngle mrked y. ii Give reson for your nswer. D 55 B y 75 C Digrm OT urtely drwn E (1384 ovemer 1996) 21 C BC B is prllel to DC ngle BC 52 Work out the vlue of i p ii q D The ngles mrked p nd r re equl. r q 52 Wht geometril nme is given to C B this type of equl ngles? (1384 ovemer 1997) p Digrm OT urtely drwn 22 The digrm represents the positions of Wign nd Mnhester. Mesure nd write down the ering of Mnhester from Wign. Find the ering of Wign from Mnhester. Wign (1385 June 1998) Mnhester 23 Mesure the ering of from B. B (1388 Mrh 2004) 38

23 Chpter 2 review questions CHTER 2 24 Work out the ering of ii B from, ii from Digrm OT urtely drwn B (1387 ovemer 2004) 25 The digrm shows the position of eh of three uildings in town. Digrm OT urtely drwn Hospitl Cinem 72 rt gllery The ering of the Hospitl from the rt gllery is 072. The Cinem is due est of the Hospitl. The distne from the Hospitl to the rt gllery is equl to the distne from the Hospitl to the Cinem. Work out the ering of the Cinem from the rt gllery. (1387 ovemer 2003) 39

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