Measurement and geometry

Transcription

1 Mesurement nd geometry 4 Geometry Geometry is everywhere. Angles, prllel lines, tringles nd qudrilterls n e found ll round us, in our homes, on trnsport, in onstrution, rt nd nture. This sene from Munih irport in Germny shows the importne of ngles, lines nd shpes in rhiteture nd design.

2 N E W C E N T U R Y M AT H S for the A n Chpter outline 4-01 Angle geometry 4-02 Angles on prllel lines 4-03 Line nd rottionl symmetry 4-04 Clssifying tringles 4-05 Clssifying qudrilterls 4-06 Properties of qudrilterls 4-07 Angle sums of tringles nd qudrilterls 4-08 Extension: Angle sum of polygon ustrlin Curriulum 8 n Wordnk Profiieny strnds U F PS R C U F PS R C ngle sum The totl of the sizes of the ngles in shpe, suh s tringle U U F F R C C onvex qudrilterl A qudrilterl whose verties ll point outwrds. U F R C digonl An intervl joining two non-djent verties of shpe U F R C exterior ngle of tringle An outside ngle of tringle formed fter extending one of the sides of the tringle U F PS R U F PS R iset To ut in hlf slene tringle A tringle with no equl sides supplementry ngles Two ngles tht dd to 180

3 Chpter Geometry n In this hpter you will: investigte ngles on stright line, ngles t point nd vertilly opposite ngles nd use results to find unknown ngles identify orresponding, lternte nd o-interior ngles when two prllel lines re rossed y trnsversl, nd the reltionships etween them investigte onditions for two lines to e prllel nd solve simple numeril prolems using resoning identify line nd rottionl symmetries lssify tringles ording to their side nd ngle properties distinguish etween onvex nd non-onvex qudrilterls desrie squres, retngles, rhomuses, prllelogrms, kites nd trpeziums pply the ngle sum of tringle nd qudrilterl nd tht ny exterior ngle of tringle equls the sum of the two interior opposite ngles SkillChek Worksheet StrtUp ssignment 4 MAT08MGWK10027 Skillsheet Types of ngles MAT08MGSS Drw two different exmples of eh type of ngle. ute ngle right ngle otuse ngle d reflex ngle 2 Clssify eh ngle size elow s eing ute, otuse, right, reflex, stright or revolution d 128 e 90 f 360 g 286 h Nme eh type of ngle(s) mrked. 4 Find the vlue of x in eh eqution. x þ 30 ¼ 90 x þ 57 ¼ 180 x þ 121 þ 77 ¼ Copy eh shpe, nme it nd drw ll xes of symmetry

4 NEW CENTURY MATHS for the Austrlin Curriulum8 d 6 Write the order of rottionl symmetry of eh shpe in question 5. 7 Whih qudrilterl hs ll four sides equl nd ll four ngles equl? Selet the orret nswer A, B, C or D. A prllelogrm B rhomus C retngle D squre 8 Drw slene tringle. 9 Drw prllelogrm nd its digonls. Are the lengths of the digonls you drew in prt equl? 4-01 Angle geometry Clssifying ngles Right ngle Stright ngle Revolution 90 (qurter-turn) 180 (hlf-turn) 360 (omplete turn) Skillsheet Strting GeoGer MAT08MGSS10019 Skillsheet Strting Geometer s Skethpd MAT08MGSS10020 Puzzle sheet Angles: A dog dy MAT08MGPS00020 Aute ngle Otuse ngle Reflex ngle Less thn 90 Between 90 nd 180 Between 180 nd 360 Worksheet Stright ngles, right ngles nd revolutions MAT08MGWK

5 Chpter Geometry Angle fts Adjent ngles Complementry ngles Angles in right ngle Angles next to eh other, Angles tht hve sum of 90, Are omplementry. shring ommon rm. for exmple, 35 nd 55. þ ¼ 90 \ABD nd \DBC re djent. A B C D Supplementry ngles Angles tht hve sum of 180, for exmple, 140 nd 40. Angles on stright line Angles t point Vertilly opposite ngles Are supplementry. x þ y ¼ 180 (In revolution) Add up to 360. Are equl. w ¼ y nd x ¼ z þ þ þ d ¼ 360 y d w z y Exmple 1 Find the vlue of eh pronumerl, giving resons y Solution x þ 70 ¼ 90 ðangles in right ngle.þ x ¼ ¼ 20 The reson is written inside rkets

6 NEW CENTURY MATHS for the Austrlin Curriulum8 ¼ 55 (Vertilly opposite ngles.) þ 55 ¼ 180 (Angles on stright line.) ¼ ¼ 125 y þ 72 þ 134 þ 90 ¼ 360 (Angles t point.) y þ 296 ¼ 360 y ¼ ¼ 64 Exerise 4-01 Angle geometry 1 Clssify eh type of ngle(s). d e f g h i 2 For eh ngle size, find the omplementry ngle d 2 3 For eh ngle size, find the supplementry ngle d 48 4 Wht is the sum of the ngles t point? Selet the orret nswer A, B, C or D. A 90 B 180 C 270 D Find the vlue of eh pronumerl, giving resons. See Exmple d y 75 e 161 f

7 Chpter Geometry g h i j 45 k n l k m n o w 118 q y 92 p p p q p 28 r Worked solutions 6 Find the vlue of eh pronumerl, giving resons. Exerise 4-01 MAT08MGWS m m 80 d 100 e f 45 g h p p m m m

8 NEW CENTURY MATHS for the Austrlin Curriulum8 7 ABC is stright line nd EB isets \DBF. Whih of the following is the size of \EBC? Selet the orret nswer A, B, C or D. A 95 B 110 C 85 D 140 D A 40 B 30 C F E 4-02 Angles on prllel lines When prllel lines re rossed y nother line (lled trnsversl), speil pirs of ngles re formed. Corresponding ngles Alternte ngles Co-interior ngles Corresponding ngles on prllel lines re equl. Alternte ngles on prllel lines re equl. Co-interior ngles on prllel lines re supplementry (dd to 180 ). Worksheet Finding the unknown ngle 1 MAT08MGWK10028 Skillsheet Angles nd prllel lines MAT08MGSS10016 Homework sheet Angle geometry MAT08MGHS10030 Worksheet Angles in prllel lines MAT08MGWK00039 Puzzle sheet Angles in prllel lines MAT08MGPS00021 Corresponding ngles re in mthing positions on the sme side of the trnsversl: orresponding mens mthing Alternte ngles re etween the prllel lines on opposite sides of the trnsversl: lternte mens going k nd forth in turns Co-interior ngles re etween the prllel lines on the sme side of the trnsversl: o-interior mens together inside

9 Chpter Geometry Exmple 2 Find the vlue of eh pronumerl, giving resons. y Solution y ¼ 62 (Alternte ngles on prllel lines) þ 110 ¼ 180 (Co-interior ngles on prllel lines) ¼ ¼ 70 Exmple 3 Prove tht the lines AB nd CD re prllel. A E 76 B C 76 F D Solution \AEF nd \EFD re lternte ngles. \AEF ¼ \EFD ¼ 76 [ AB CD [ AB CD mens Therefore (Alternte ngles re equl) line AB is prllel to line CD Exerise 4-02 Angles on prllel lines 1 Is eh mrked pir of ngles orresponding, lternte or o-interior? d e f

10 NEW CENTURY MATHS for the Austrlin Curriulum8 2 Whih ngle is orresponding to the ngle mrked g? Selet A, B, C or D. B A C D f g e h 3 For the digrm in question 2, nme n ngle tht is: o-interior to D lternte to the ngle mrked e equl to C d orresponding to B e supplementry to the ngle mrked e. 4 Find the vlue of the pronumerl in eh digrm, giving resons for your nswers d 61 See Exmple 2 d e f 120 p m m 5 Find the vlue of eh pronumerl, giving resons. Worked solutions Exerise d MAT08MGWS10029 d e f g h m q 80 p g h i 72 y n y w

11 Chpter Geometry 6 Wht is the vlue of y in this digrm? Selet the orret nswer A, B, C or D. A 85 B 40 C 35 D y See Exmple 3 7 For eh digrm, deide whether AB is prllel to CD. If it is, then prove it. A A C d E F A E 48 C B 100 E 120 B C F 58 F D D B D A 115 E B C 65 F D Worksheet Symmetry 4-03 Line nd rottionl symmetry MAT08MGWK10029 Skillsheet Line nd rottionl symmetry MAT08MGSS10017 A shpe is symmetril if it looks the sme fter it hs undergone hnge of position or movement. The two types of symmetry re line symmetry nd rottionl symmetry. When shpe with line symmetry is folded long line, lled n xis of symmetry, the two hlves of the shpe fit extly on top of eh other. One hlf is the refletion or mirror-imge of the other hlf. xis of symmetry This shpe hs one xis of symmetry. When shpe with rottionl symmetry is rotted (spun) out point, lled the entre of symmetry, it fits extly on itself t lest one efore one full revolution (360 ). The numer of times the shpe fits on itself in one revolution is lled its order of rottionl symmetry. entre of symmetry This shpe hs rottionl symmetry of order

12 NEW CENTURY MATHS for the Austrlin Curriulum8 Exerise 4-03 Line nd rottionl symmetry 1 Copy eh shpe nd mrk its xes of symmetry. d e f 2 Count the numer of xes of symmetry of eh shpe. d 3 Deide whether eh shpe hs rottionl symmetry. If it does, stte the order of rottionl symmetry. d 4 Copy the pitl letters elow tht hve line symmetry nd drw their xes of symmetry. D H I K M N O R S V X Z Copy the pitl letters ove tht hve rottionl symmetry nd mrk their entre of symmetry. 5 Drw shpe or pttern tht hs: one xis of symmetry two xes of symmetry four xes of symmetry 6 Drw shpe or pttern tht hs rottionl symmetry: of order 2 of order

13 Chpter Geometry 7 For eh shpe: i find how mny xes of symmetry it hs ii stte whether it hs rottionl symmetry nd if it does, stte the order d e f 8 Wht shpe hs: n infinite numer of xes of symmetry? n infinite order of rottionl symmetry? Worksheet Properties of tringles MAT08MGWK10030 Worksheet Tringle geometry MAT08MGWK10031 Skillsheet Nming shpes MAT08MGSS10018 Homework sheet 4-04 Clssifying tringles Tringles n e lssified in two wys: y their sides or y their ngles. Clssifying y sides Equilterl tringle Isoseles tringle Slene tringle Three equl sides (Also three equl ngles, eh 60 ) Two equl sides (Also two equl ngles, opposite the equl sides) No equl sides (Also no equl ngles) 60 Symmetry nd tringles MAT08MGHS10031 Puzzle sheet Clssifying tringles MAT08MGPS Clssifying y ngles Aute-ngled tringle Otuse-ngled tringle Right-ngled tringle Three ute ngles (less thn 90 ) One otuse ngle (etween 90 nd 180 ) One right ngle (90 )

14 NEW CENTURY MATHS for the Austrlin Curriulum8 Exerise 4-04 Clssifying tringles 1 Clssify eh tringle ording to its sides nd ngles. d e f g h i j k l 2 Sketh tringle tht is: right-ngled nd isoseles equilterl slene nd otuse-ngled d ute-ngled nd slene e right-ngled nd slene f ute-ngled nd isoseles 3 Whih of the following desries this tringle when we lssify it y sides nd ngles? Selet the orret nswer A, B, C or D. A isoseles nd otuse-ngled B isoseles nd ute-ngled C slene nd otuse-ngled D slene nd ute-ngled 4 Is it possile to drw n otuse-ngled equilterl tringle? Justify your nswer. 5 Whih tringles in question 1 hve: line symmetry? rottionl symmetry? Find the vlue of eh pronumerl, giving reson m y m d l mm 4.8 mm e m 60 m f r 10 m m m

15 Chpter Geometry 7 Is it possile to drw tringle with two otuse ngles? Why? 8 Whih tringle is oth otuse-ngled nd slene? Selet A, B, C or D. A B C D 9 Copy nd omplete this tle. Tringle Equilterl tringle Isoseles tringle Slene tringle Numer of xes of symmetry Order of rottionl symmetry Investigtion: The perpendiulr isetor in n isoseles tringle nabc is n isoseles tringle with AC ¼ AB. It hs one xis of symmetry, AD. 1 Why is CD ¼ DB? 2 Why is \ADC ¼ \ADB? 3 Wht is the size of \ADC nd \ADB? 4 AD isets side CB. Wht does iset men? 5 AD CB. Wht does men? 6 In n isoseles tringle, the xis of symmetry is the perpendiulr isetor of the uneven side. Explin wht this mens in your own words. C A D B Mentl skills 4A Mths without lultors Converting frtions nd deimls to perentges To onvert frtion or deiml into perentge, multiply it y 100%. 1 Study eh exmple. 2 5 ¼ % ¼ % ¼ % ¼ 40% ¼ % ¼ 3 100% ¼ % ¼ % ¼ 60%

16 NEW CENTURY MATHS for the Austrlin Curriulum8 2 Now onvert eh frtion to perentge f 30 g h 4 5 k l m Study eh exmple. 0:41 ¼ 0: % ¼ 0:41 ¼ 0:08 ¼ 41% ¼ 8% 0:9 ¼ 0: % ¼ 0:90 ¼ 90% 4 Now onvert eh deiml to perentge. d i 20 n :08 ¼ 0: % d 0:375 ¼ 0: % ¼ 0:375 ¼ 37:5% e j 80 o d 0.6 e 0.1 f g 0.59 h i 0.84 j 0.7 k l m 0.91 n o Just for the reord It s ll Greek or Ltin to me! Mny of our words in geometry ome from Greek or Ltin. Ltin ws the lnguge of the nient Romn Empire. Word Origin Mening Equilterl Ltin: equus ltus Equl sides Equingulr Ltin: equus ngulus Equl orners Isoseles Greek: isos skelos Equl legs Slene Greek: sklenos Uneven leg Aute Ltin: utus Shrp Otuse Ltin: otusus Dull or lunt Reflex Ltin: reflexus Bent k Tringle Ltin: tri ngulus Three orners Retngle Ltin: retus ngulus Right orners Qudrilterl Ltin: qudri ltus Four sides Polygon Greek: poly gonon Mny ngles Digonl Greek: di gonios From ngle to ngle Trpezium/Trpezoid Ltin/Greek: trpez Smll tle Explin wht this sentene mens, nd illustrte with digrm: A rhomus is equilterl ut not equingulr

17 Chpter Geometry Worksheet Properties of qudrilterls MAT08MGWK Clssifying qudrilterls A qudrilterl is ny shpe with four stright sides. A qudrilterl my e onvex or non-onvex. Worksheet Clssifying qudrilterls MAT08MGWK10033 Convex qudrilterl Non-onvex qudrilterl Worksheet Alwys, sometimes, never true MAT08MGWK10034 Puzzle sheet Wht shpe m I? MAT08MGPS10013 Video tutoril All verties (orners) point outwrds. All digonls lie within the shpe. All ngles re less thn 180. There re six speil types of qudrilterls. One vertex points inwrds. One digonl lies outside the shpe. One ngle is more thn 180 (reflex ngle). Clssifying qudrilterls Trpezium Prllelogrm Retngle MAT08MGVT00002 Puzzle sheet Singing in the r MAT08MGPS00004 Worksheet Clssifying tringles nd qudrilterls One pir of prllel sides Two pirs of prllel sides Four right ngles Rhomus Squre Kite MAT08MGWK00010 Four equl sides Four equl sides nd four right ngles Two pirs of equl djent sides Adjent mens next to eh other. Exerise 4-05 Clssifying qudrilterls 1 Is eh qudrilterl onvex or non-onvex?

18 NEW CENTURY MATHS for the Austrlin Curriulum8 d e f 2 Nme eh qudrilterl. d 3 Nme ll the qudrilterls tht hve: four right ngles extly one pir of prllel sides four equl sides d opposite sides equl e opposite sides prllel f two pirs of equl djent sides 4 Whih qudrilterl is lso lled dimond? 5 Copy nd omplete this tle. Qudrilterl Retngle Prllelogrm Trpezium Rhomus Squre Kite Numer of xes of symmetry Order of rottionl symmetry 6 In the digrm on the right, wht type of qudrilterl is: ACDF? FBCD? F A B 7 A prllelogrm is ny qudrilterl with oth pirs of opposite sides prllel. Whih of the following is not speil type of prllelogrm? Selet the orret nswer A, B, C or D. A squre B kite C retngle D rhomus 8 Sketh eh of these qudrilterls, showing its min fetures. retngle trpezium rhomus d kite 9 Whih of the following sttements re lwys true? (Explin your nswers) A rhomus is squre. A squre is rhomus. A retngle is prllelogrm. d A prllelogrm is qudrilterl with its opposite sides prllel nd equl. e The digonls of prllelogrm meet t right ngles. f A squre is retngle. g A retngle is squre. E D C

19 Chpter Geometry Tehnology Properties of qudrilterls Tehnology GeoGer: Mking qudrilterls This tivity will use GeoGer to onstrut qudrilterls. For eh qudrilterl, set up your drwing pge y liking View. Mke sure tht Grid is tiked nd Axes is not tiked. MAT08MGTC00005 Tehnology GeoGer: Qudrilterl sides nd ngles MAT08MGTC00002 Tehnology GeoGer: Qudrilterl digonls MAT08MGTC00003 Squre 1 To drw squre, lik on Regulr Polygon. Use the grid to drw n intervl of length 7.5 m. Selet 4 points. Mke sure tht the lels re showing. 2 Use intervl etween two points to onstrut the two digonls for the squre. Selet distne or length to mesure the length of eh digonl in the squre. Wht do you notie? 3 Now mesure the Angle of eh vertex of the squre. Wht do you notie? 4 Drw nother squre with side length 9 m. Repet steps 2 nd 3. List two properties of the squre. Retngle 1 To drw retngle, lik on Polygon. Use the grid to drw retngle with sides 6 m y 3 m. 2 Use distne or length to mesure eh side length of the retngle. 3 Now mesure the Angle of eh vertex to hek your ury. Corret ny inurte sides using Move. Wht symol is shown on eh vertex when the ngle is shown s extly 90? 4 Drw nother retngle with length 5 m nd width 8.4 m. Use the instrutions given in step 2 ove to mesure the side lengths nd ngles nd lso to orret ny inurte sides nd/or ngles. 5 Use intervl etween two points to onstrut the two digonls for eh of your retngles. Selet distne or length to mesure the length of eh digonl in every retngle. Wht do you notie? 6 Complete this property: The in retngle re. 7 In one retngle, selet Interset two ojets nd lik on the two digonls. Use distne or length to mesure the distne from the intersetion point to the vertex for eh digonl. Repet for the seond retngle. Wht do you notie? 8 Complete this property: The of retngle eh other

20 NEW CENTURY MATHS for the Austrlin Curriulum8 Prllelogrm 1 Selet Intervl with given length from point. Mke the intervl 6 m long. Clik Show lel. 2 Now selet Intervl with given length from point nd lik point A. Mke the intervl 4 m. Use Move nd drg the new intervl s shown elow. Lel the new point, C. 3 Clik Prllel line nd selet line AB nd point C. Now selet AC nd point B. 4 Use Interset two ojets to rete the missing vertex of the prllelogrm. Lel the vertex. 5 Use distne or length to show the length of eh side of the prllelogrm. Wht do you notie? 6 Complete the following: The sides of prllelogrm re. 7 Now use Angle to find the size of \CAB nd \CDB. Repet for \ACD nd \ABD. Wht do you notie? 8 Complete the following: The ngles of prllelogrm re. 9 Now drw the digonls of the prllelogrm. Use distne or length to show the length of eh digonl. Wht do you notie? 10 Complete the following: The digonls of prllelogrm re. 11 Do the digonls of prllelogrm iset eh other? Repet step 7 from Retngle to help you. 12 Complete the following: The digonls of prllelogrm iset eh other. 13 Drg ny verties of the prllelogrm tht you n. Is it possile to drw other prllelogrms with the sme dimensions, 6 m y 4 m? Wht do you notie? 14 Use your GeoGer skills to urtely onstrut other qudrilterls suh s rhomus, kite or trpezium Properties of qudrilterls Exerise 4-06 Properties of qudrilterls 1 Copy the tle elow or use the link to print one out. Aurtely drw eh of the six speil qudrilterls elow on sheet of pper, then ut them out. Use your shpes to help you omplete the tle of qudrilterl properties. Worksheet Properties of qudrilterls MAT08MGWK10032 Trpezium One pir of opposite sides re

21 Chpter Geometry Kite Two pirs of djent sides re One pir of opposite re equl. Digonls interset t Prllelogrm d Rhomus e Retngle f Squre sides re equl. Opposite re prllel. Opposite ngles re Digonls eh other. All re equl. sides re ngles re Digonls iset eh other t ngles. Digonls the ngles of the rhomus. Opposite sides re Opposite sides re lso All ngles re Digonls re Digonls eh other. All sides re All ngles re Opposite re prllel. Digonls re Digonls iset eh other t 2 Whih qudrilterls hve eh property? Opposite sides re equl Digonls ross t right ngles Opposite ngles re equl d One pir of opposite sides re prllel e Digonls iset eh other f Opposite sides re prllel g Adjent sides re of different lengths h Two equl digonls i All ngles re equl j All sides re equl 3 Copy nd fill in the lnks to find the vlues of nd. 70 þ 70 ¼ 180 ( ngles on lines) ¼ ¼ (opposite of prllelogrm)

22 NEW CENTURY MATHS for the Austrlin Curriulum8 4 Find the vlue of eh pronumerl. Worked solutions m y Exerise 4-06 MAT08MGWS d e 41 f 87 l y g 105 h i m n 33 5 m 7 m j 7 m 4 m k I m qudrilterl with opposite sides equl nd prllel. My digonls re equl nd I hve two xes of symmetry. Whih qudrilterl m I? Selet the orret nswer A, B, C or D. A prllelogrm B retngle C squre D rhomus 6 I m qudrilterl with opposite sides equl. My digonls iset eh other nd meet t right ngles. Whih qudrilterl m I? Selet A, B, C or D. A prllelogrm B trpezium C retngle D rhomus 7 A retngle is qudrilterl with four right ngles. Whih one of the following is speil type of retngle? Selet A, B, C or D. A squre B kite C prllelogrm D rhomus 8 A rhomus is qudrilterl with ll sides equl. Whih one of the following is speil type of rhomus? Selet A, B, C or D. A squre B kite C prllelogrm D retngle

23 Chpter Geometry Tehnology Exterior ngle of tringle In this tivity we will use GeoGer to disover n importnt property out the interior nd exterior ngles of ny tringle. To set up your drwing pge, lik View nd mke sure tht Grid is seleted nd Axes is not seleted. 1 Drw tringle using the polygon tool. Right-lik on eh vertex nd selet Show lel if no lels re showing. 2 Selet Ry through two points from the third ion menu nd drw ry from A through C. 3 Selet New Point from the seond ion menu. Insert the new point on ry AC outside intervl AC. Show lel D. 4 Use Angle from the eighth ion menu to find the size of \ABC, \BAC nd \BCD. 5 Clulte \ABC þ \BAC. 6 Compre your nswer from question 5 with the size of \BCD. Wht do you notie? 7 Copy nd omplete: The ngle of tringle is to the sum of the opposite ngles

24 NEW CENTURY MATHS for the Austrlin Curriulum Angle sums of tringles nd qudrilterls Worksheet Find the unknown ngle 2 MAT08MGWK10035 Angle sum of tringle Summry The ngle sum of tringle is 180 þ þ ¼ 180 Video tutoril Angle sums of tringles nd qudrilterls MAT08MGVT10007 Video tutoril Angle in polygons MAT08MGVT00006 Homework sheet Qudrilterls nd ngle sums Exmple 4 Find the vlue of eh pronumerl, giving resons y Solution x þ 42 þ 39 ¼ 180 (Angle sum of tringle) x ¼ ¼ 99 y þ 67 þ 67 ¼ 180 (Angle sum of n isoseles tringle) y ¼ ¼ MAT08MGHS10032 Quiz Shpes nd ngles MAT08MGQZ00006 Puzzle sheet Mixed ngles MAT08MGPS00023 Puzzle sheet Angles in tringles MAT08MGPS00022 The exterior ngle of tringle Summry Tehnology GeoGer: Exterior ngle of tringle MAT08MGTC00007 The exterior ngle of tringle is equl to the sum of the two interior opposite ngles. z ¼ x þ y y z Worksheet Angles in tringles MAT08MGWK

25 Chpter Geometry Exmple 5 Find the vlue of eh pronumerl, giving resons. 31 k 83 m Solution m ¼ 83 þ 31 (Exterior ngle of tringle) ¼ 114 k þ 51 ¼ 112 (Exterior ngle of tringle) k ¼ ¼ 61 Angle sum of qudrilterl Any qudrilterl n e divided into two tringles long one of its digonls. Beuse the ngles in eh tringle dd to 180, the ngles in oth tringles dd to ¼ 360. u þ v þ w ¼ 180 nd þ y þ z ¼ 180 ) Angle sum of qudrilterl ¼ 180 þ 180 ¼ 360 u v w z y Summry The ngle sum of qudrilterl is 360. þ þ þ d ¼ 360 d This property is true for oth onvex nd non-onvex qudrilterls

26 NEW CENTURY MATHS for the Austrlin Curriulum8 Exmple 6 Find the vlue of eh pronumerl, giving resons q 230 k Solution q þ 91 þ 79 þ 68 ¼ 360 (Angle sum of qudrilterl) q þ 238 ¼ 360 q ¼ ¼ 122 k þ 25 þ 37 þ 230 ¼ 360 (Angle sum of qudrilterl) k þ 292 ¼ 360 k ¼ ¼ 68 Exerise 4-07 Angle sums of tringles nd qudrilterls 1 Find the vlue of eh pronumerl d e f d e f See Exmple 4 g g h i 30 h i

27 Chpter Geometry See Exmple 5 2 Find the vlue of eh pronumerl d e f d e f g h i Worked solution h Exerise 4-07 MAT08MGWS10031 See Exmple 6 g i 68 3 Find the vlue of eh pronumerl d e f d e f 54 g 115 h i 72 h 83 i g 4 Whih of the following is the vlue of u? Selet the orret nswer A, B, C or D. 50 A 50 B 60 C 70 D u

28 NEW CENTURY MATHS for the Austrlin Curriulum8 5 Whih of the following is the vlue of r? Selet A, B, C or D. A 25 B 35 C 50 D r 6 Whih eqution is orret for the tringle on the right? Selet A, B or C. A n ¼ l þ m B n ¼ k þ m C n ¼ l þ k n k m l 7 Whih of the following is the vlue of x? Selet A, B, C or D. A 115 B 65 C 127 D Investigtion: Angle sum of polygon We know tht the ngle sum of tringle is 180 nd tht the ngle sum of qudrilterl is 360, ut how do we find the ngle sum of other onvex polygons? A hexgon n e divided into four tringles y drwing the digonls from one vertex The sum of the ngles in hexgon ¼ðngle sum of tringleþ 3 4 ¼ ¼ 720 An otgon n e divided into six tringles y drwing the digonls from one vertex. The totl of the ngles in n otgon ¼ ¼ How is the numer of sides relted to the numer 5 of tringles formed in shpe? Copy nd omplete the following sentenes. The ngle sum of polygon with n sides is A ¼ (numer of sides ) or: A ¼ (n ) 3 Find the ngle sum of eh polygon. 11-sided polygon 20-sided polygon 14-sided polygon

29 Chpter Geometry Worksheet Angle sum of polygon 4-08 Extension: Angle sum of polygon MAT08MGWK10036 Tehnology worksheet Angle sum of polygon MAT08MGCT10002 A onvex polygon hs ll verties pointing outwrds. A regulr polygon hs ll sides the sme length nd ll ngles the sme size. Exmple 7 These shpes re ll hexgons (six-sided). Whih hexgons re onvex? Whih hexgon is regulr? i ii iii Solution Hexgons ii nd iii re onvex euse ll of their verties point outwrds. Hexgon iii is regulr, euse ll its sides re equl nd ll its ngles re equl. Summry The ngle sum of polygon with n sides is given y the formul A ¼ 180(n 2). This property pplies to oth onvex nd non-onvex polygons. Exmple 8 Worksheet Angles in regulr polygons MAT08MGWK00041 Find the size of one ngle in regulr hexgon. Solution A hexgon hs six sides (n ¼ 6). Angle sum of hexgon ¼ 180ð6 2Þ ¼ ¼ 720 For regulr hexgon: one ngle ¼ ¼ 120 [ Eh ngle in regulr hexgon is

30 NEW CENTURY MATHS for the Austrlin Curriulum8 Exerise 4-08 Angle sum of polygon 1 Copy these polygons nd, eneth eh drwing, write their orret nme (hosen from the list). Sy if eh polygon is regulr or irregulr, onvex or non-onvex. hexgon nongon heptgon degon otgon tringle qudrilterl pentgon See Exmple 7 d e f g h 2 Copy nd omplete this tle. Polygon Numer of sides Sum of ngles inside polygon hexgon heptgon otgon nongon degon 3 Find the sum of the interior ngles in: 15-gon 20-gon 25-gon d 100-gon 4 Find the vlue of eh pronumerl d d d d d

31 Chpter Geometry d e g f g e e e e e e 50 See Exmple 8 Worked solution Exerise 4-08 e e 5 Find the size of one interior ngle in eh regulr polygon. squre equilterl tringle regulr hexgon d regulr otgon e regulr degon f regulr pentgon g regulr dodegon 6 Find the numer of sides of the polygon tht hs n ngle sum of: d 9180 e MAT08MGWS10032 Mentl skills 4B Mths without lultors Converting deimls nd perentges to frtions 1 Consider eh of the following exmples. 0:35 ¼ ¼ 7 20 (two deiml ples, two zeros in the denomintor) 0:8 ¼ ¼ 4 5 (one deiml ple, one zero in the denomintor) 0:64 ¼ ¼ d 0:22 ¼ ¼ Now onvert eh deiml to frtion d 0.14 e 0.06 f 0.85 g 0.32 h 0.49 i 0.56 j 0.9 k 0.72 l 0.65 m 0.2 n 0.24 o Consider eh of the following exmples. 26% ¼ ¼ 13 40% ¼ ¼ % ¼ ¼ 2 25 d 95% ¼ ¼

32 NEW CENTURY MATHS for the Austrlin Curriulum8 4 Now onvert eh perentge to frtion. 76% 10% 80% d 45% e 88% f 56% g 75% h 31% i 68% j 5% k 60% l 54% m 6% n 49% o 82% Power plus As questions eome more omplex, it my not e possile to find the nswer in one step. It my e neessry to find nother ngle first. 1 Find the vlue of x in eh digrm. (Give resons for ll steps.) d 40 e f 6 m 36 x m g 46 h 25 i j k l

33 Chpter 4 review n Lnguge of mths Puzzle sheet Geometry rossword MAT08MGPS10014 ute lternte ngle sum xis/xes o-interior omplementry onvex orresponding digonl equilterl exterior ngle interior ngle isoseles kite otuse prllelogrm qudrilterl reflex rhomus rottionl symmetry slene supplementry trpezium vertilly opposite Worksheet Mind mp: Geometry MAT08MGWK Wht re supplementry ngles? 2 Nme the types of ngles ssoited with prllel lines ut y trnsversl. 3 Wht type of tringle hs one ngle tht is greter thn 90? 4 Wht is the ngle sum of qudrilterl? 5 Illustrte the differene etween n interior ngle nd n exterior ngle of tringle. 6 Wht does equilterl men? Wht is the ommon nme for n equilterl prllelogrm? n Topi overview Do you think this hpter is useful? Why? Wht did you lern in this hpter? How onfident do you feel with geometry? List nything in this hpter tht you did not understnd. Show your teher. Copy nd omplete this mind mp of the topi, dding detil to its rnhes nd using pitures, symols nd olour where needed. Ask your teher to hek your work. Angles nd prllel lines Angle geometry Angles Types of ngles Properties Tringles Angle sum Types of tri ngles Exterior ngle Qudrilterls GEOMETRY Types of qudrilterls Properties Angle sum Line nd rottionl symmetry

34 Chpter 4 revision 1 Nme eh type of ngle. See Exerise Find the vlue of eh pronumerl, giving resons. See Exerise y 85 d e n 72 w Nme eh type of ngle pir. See Exerise Find the vlue of eh pronumerl, giving resons. m See Exerise 4-02 d 100 y 88 m p

35 Chpter 4 revision See Exerise Copy eh shpe nd mrk in its xes of symmetry. See Exerise 4-03 See Exerise For eh shpe in question 5, deide whether or not it hs rottionl symmetry. If it does, stte the order of rottionl symmetry. 7 Clssify eh of these tringles y its sides nd ngles. d e f g h i See Exerise 4-05 See Exerise 4-06 See Exerise Drw net digrm of eh of the following qudrilterls. prllelogrm kite trpezium 9 Nme ll the qudrilterls tht hve the following properties. ll sides equl in length no prllel sides ll ngles right ngles d two pirs of opposite ngles equl e digonls iset eh other 10 Find the vlue of eh pronumerl y d y e f m

36 Chpter 4 revision 11 Find the vlue of eh pronumerl. See Exerise y m d 60 d 130 e 50 f 56 g Find the vlue of eh pronumerl m 120 y See Exerise