mesurement mesurement n n geometry geometry UNORRETE PGE PROOFS topi 10 edutive geometry 10.1 Overview Why lern this? Lerning out geometry inludes eing le to reson dedutively nd to prove logilly tht ertin mthemtil sttements re true. It is importnt to e le to prove theories metiulously nd step y step in order to show tht the onlusions rehed re soundly sed. Mthemtiins spend most of their time trying to prove new theories, nd they rely hevily on ll the proofs tht hve gone efore. Resoning skills, nd hene the ility to prove theories, n e developed nd lerned through prtie nd pplition. Wht do you know? 1 think List wht you know out geometry. Use thinking tool suh s onept mp to show your list. 2 PIr Shre wht you know with prtner nd then with smll group. 3 shre s lss, rete thinking tool suh s lrge onept mp tht shows your lss s knowledge of geometry. Lerning sequene 10.1 10.2 10.3 10.4 10.5 10.6 Overview ngles, tringles nd ongruene Similr tringles Qudrilterls Polygons Review ONLINE ONLY 10edutiveGeometry.indd 394 8/20/14 6:30 PM
mesurement n geometry UNORRETE PGE PROOFS WtH this VIeo The story of mthemtis: serhlight I: eles-1849
mesurement N geometry UNORRETE PGE PROOFS 10.2 ngles, tringles nd ongruene Eulid (. 300 ) ws the mthemtiin who developed systemti pproh to geometry, now referred to s Euliden geometry, tht relied on mthemtil proofs. proof is n rgument tht shows why sttement is true. theorem is sttement tht n e demonstrted to e true. To demonstrte tht sttement is proven, forml lnguge needs to e used. It is onventionl to use the following struture when setting out theorem. Given: summry of the informtion given To prove: sttement tht needs to e proven onstrution: desription of ny dditions to the digrm given Proof: sequene of steps tht n e justified nd form prt of forml mthemtil proof. ngles ngles t point The sum of the ngles t point is 360. + + + d + e = 360 Supplementry ngles The sum of the ngles on stright line is 180. ngles, nd re supplementry ngles. + + = 180 Vertilly opposite ngles Theorem 1: Vertilly opposite ngles re equl. Given: Stright lines nd interset t O. To prove: O = O nd O = O onstrution: Lel O s, O s nd O s. Proof: Let O =, O = nd O =. + = 180 + = 180 + = + = So, O = O. Similrly, O = O. d O e (supplementry ngles) (supplementry ngles) + + + d + e = 360 + + = 180 396 Mths Quest 10 + 10
mesurement N geometry UNORRETE PGE PROOFS Prllel lines If two lines re prllel nd ut y trnsversl, then: o interior ngles re equl. For exmple, =. orresponding ngles re equl. For exmple, = d. lternte ngles re equl. For exmple, =. opposite ngles re equl. For exmple, = d. ngle properties of tringles Theorem 2 Theorem 2: The sum of the interior ngles of tringle is 180. Given: Δ with interior ngles, nd To prove: + + = 180 onstrution: rw line prllel to, pssing through nd lel it E s shown. Lel s x nd E s y. Proof: = x (lternte ngles) = y (lternte ngles) x + + y = 180 + + = 180 x y E (supplementry ngles) Equilterl tringles It follows from Theorem 2 tht eh interior ngle of n equilterl tringle is 60, nd, onversely, if the three ngles of tringle re equl, then the tringle is equingulr. + + = 180 3 = 180 = 60 Trnsversl (sum of interior ngles in tringle is 180 ) d Topi 10 edutive geometry 397
mesurement N geometry Theorem 3: The exterior ngle of tringle is equl to the sum of the opposite interior ngles. UNORRETE PGE PROOFS Given: Δ with the exterior ngle lelled d To prove: d = + Proof: + d = 180 (supplementry ngles) + + = 180 (sum of interior ngles in tringle is 180 ) d = + ongruent tringles ongruent tringles hve the sme size nd the sme shpe; tht is, they re identil in ll respets. The symol used for ongrueny is. For exmple, Δ in the digrm elow is ongruent to ΔPQR. This is written s Δ ΔPQR. Note tht the verties of the two tringles re written in orresponding order. There re five tests designed to hek whether tringles re ongruent. The tests re summrised in the tle elow. Test igrm revition ll three sides in one tringle re equl in length to the orresponding sides in the other tringle. Two orresponding sides nd the inluded ngle re the sme in oth tringles. Two orresponding ngles nd pir of orresponding sides re the sme in oth tringles. pir of orresponding ngles nd non-inluded side re equl in oth tringles. P Q R d SSS SS S S 398 Mths Quest 10 + 10
mesurement n geometry UNORRETE PGE PROOFS Test igrm revition The hypotenuse nd one pir of the other orresponding sides in two right-ngled tringles re the sme in two right-ngled tringles. RHS In eh of the tests we need to show three equl mesurements out pir of tringles in order to show they re ongruent. WorKe exmple 1 Selet pir of ongruent tringles from the digrms elow, giving reson for your nswer. think 50 15 m 95 1 In eh tringle the length of the side opposite the 95 ngle is given. If tringles re to e ongruent, the sides opposite the ngles of equl size must e equl in length. rw your onlusion. 2 To test whether Δ is ongruent to ΔPQR, first find the ngle. 3 pply test for ongruene. Tringles nd PQR hve pir of orresponding sides equl in length nd 2 pirs of ngles the sme, so drw your onlusion. Isoseles tringles tringle is isoseles if the lengths of two sides re equl, ut the third side is not equl. Theorem 4 Theorem 4: The ngles t the se of n isoseles tringle re equl. Q 95 18 m N L 35 35 P R 95 15 m M WrIte ll three tringles hve equl ngles, ut the sides opposite the ngle 95 re not equl. = PR = 15 nd LN = 18 m. Δ: = 50, = 95, = 180 50 95 = 35 pir of orresponding ngles ( = Q nd = R) nd non inluded side (P = PR) re equl. Δ ΔPQR (S) Topi 10 edutive geometry 399
mesurement n geometry UNORRETE PGE PROOFS Given: = To prove: = onstrution: rw line from the vertex to the midpoint of the se nd lel the midpoint. is the isetor of. Proof: In Δ nd Δ, = = = Δ Δ = d (ommon side) (onstrution, is the midpoint of ) (given) (SSS) onversely, if two ngles of tringle re equl, then the sides opposite those ngles re equl. It lso follows tht = = d 2d = 180 nd tht (supplementry) d = 90 WorKe exmple 2 Given tht Δ Δ, find the vlues of the pronumerls in the figure elow. think 40 x z y 3 m 1 In ongruent tringles orresponding sides re equl in length. Side (mrked x) orresponds to side, so stte the vlue of x. 2 Sine the tringles re ongruent, orresponding ngles re equl. Stte the ngles orresponding to y nd z nd hene find the vlues of these pronumerls. WrIte Δ Δ =, = x, = 3 So x = 3 m. = = 40, = y So y = 40. = = z, = 90 So z = 90. 400 Mths Quest 10 + 10
mesurement n geometry WorKe exmple 3 Prove tht ΔPQS is ongruent to ΔRSQ. P Q UNORRETE PGE PROOFS think Exerise 10.2 ngles, tringles nd ongruene InIVIuL PtHWys PrtIse Questions: 1 5, 7, 9, 11 onsolite Questions: 1 5, 6, 8 10, 12, 13 FLueny 1 etermine the vlues of the unknown in eh of the following. 56 d S Individul pthwy intertivity int-4612 58 d R WrIte 1 Write the informtion given. Given: Retngle PQRS with digonl QS. 2 Write wht needs to e proved. To prove: tht ΔPQS is ongruent to ΔRSQ. QP = SR (given) SPQ = SRQ = 90 (given) QS is ommon. 3 Selet the pproprite ongrueny test for proof. (In this se it is RHS euse the tringles hve n equl side, right ngle nd ommon hypotenuse.) So ΔPQS ΔRSQ (RHS) mster Questions: 1 14 120 30 e e 44 refletion How n you e ertin tht two fi gures re ongruent? 62 e Topi 10 edutive geometry 401
mesurement n geometry 2 WE1 Selet pir of ongruent tringles in eh of the following, giving reson for your nswer. ll side lengths re in m. 65 65 UNORRETE PGE PROOFS do 5276 do 5277 do 5280 d 3 4 3 I 65 70 40 2 I I 110 I 6 m 3 II 70 4 II 4.8 3.5 3 4 6 m 110 2 4 4.8 II 40 II 3.5 4 45 5 3 III III 3 III 110 40 6 m 3.5 III 4.8 2.5 unerstning 3 Find the missing vlues of x nd y in eh of the following digrms. Give resons for your nswers. x y 6 y E O x 130 402 Mths Quest 10 + 10
mesurement N geometry d UNORRETE PGE PROOFS x y 99 32 y x 45 4 WE2 Find the vlue of the pronumerl in eh of the following pirs of ongruent tringles. ll side lengths re in m. d 4 4 7 y 30 x RESONING 3 x e 85 x x 40 5 WE3 Prove tht eh of the following pirs of tringles re ongruent. P y n m P z x z 80 Q 30 y S R R S Q Topi 10 edutive geometry 403
mesurement N geometry UNORRETE PGE PROOFS P S Q d e Q P R R S 6 Note: There my e more thn one orret nswer. Whih of the following is ongruent to the tringle shown t right? M 3 m 5 m 35 3 m 5 m 5 m 3 m 35 35 3 m 3 m 35 5 m 35 7 5 m Prove tht Δ Δ nd hene find the vlues of the pronumerls in eh of the following. 30 30 7 m w 70 x x 30 y 65 x y 4 m 40 40 z 8 Explin why the tringles shown t right re not neessrily ongruent. 40 5 m 5 m 40 7 m 7 m 404 Mths Quest 10 + 10 10edutiveGeometry.indd 404 8/20/14 6:30 PM
mesurement N geometry 9 Explin why the tringles shown t right re not ongruent. 30 70 8 m 70 8 m 30 UNORRETE PGE PROOFS 10 Show tht ΔO ΔO, if O is the entre of the irle. 11 If = =, prove tht is right ngle. 12 If = nd = E in the digrm shown, prove tht E. E PROLEM SOLVING 13 is n isoseles tringle in whih nd re equl in length. F is right-ngled tringle. Show tht tringle EF is n isoseles tringle. 14 Tringles nd EF re ongruent. 50 (2x + y) O F E Find the vlues of x, y nd z. 110 (x + z) (3y + z) E F Topi 10 edutive geometry 405
mesurement N geometry UNORRETE PGE PROOFS 10.3 Similr tringles Similr figures Two geometri shpes re similr when one is n enlrgement or redution of the other shpe. n enlrgement inreses the length of eh side of figure in ll diretions y the sme ftor. For exmple, in the digrm shown, tringle is n enlrgement of tringle y ftor of 3 from its entre of enlrgement t O. The symol for similrity is ~ nd is red s is similr to. The imge of the originl ojet is the enlrged or redued shpe. To rete similr shpe, use sle ftor to enlrge or redue the originl shpe. O The sle ftor n e found using the formul elow nd the lengths of pir of orresponding sides. imge side length Sle ftor = ojet side length If the sle ftor is less thn 1, the imge is redued version of the originl shpe. If the sle ftor is greter thn 1, the imge is n enlrged version of the originl shpe. Similr tringles Two tringles re similr if: the ngles re equl, or the orresponding sides re proportionl. onsider the pir of similr tringles elow. 3 5 6 U 10 4 V 8 W 406 Mths Quest 10 + 10
mesurement N geometry UNORRETE PGE PROOFS The following sttements re true for these tringles. Tringle UVW is similr to tringle or, using symols, ΔUVW Δ. The orresponding ngles of the two tringles re equl in size: = WUV, = UVW nd = UWV. The orresponding sides of the two tringles re in the sme rtio. UV = VW = UW = 2; tht is, ΔUVW hs eh of its sides twie s long s the orresponding sides in Δ. The sle ftor is 2. Testing tringles for similrity Tringles n e heked for similrity using one of the tests desried in the tle elow. Test igrm revition Two ngles of tringle re equl to two ngles of nother tringle. This implies tht the third ngles re equl, s the sum of ngles in tringle is 180. The three sides of tringle re proportionl to the three sides of nother tringle. Two sides of tringle re proportionl to two sides of nother tringle, nd the inluded ngles re equl. The hypotenuse nd seond side of right ngled tringle re proportionl to the hypotenuse nd seond side of nother right ngled tringle. k k k k k k k SSS SS RHS Note: When using the equingulr test, only two orresponding ngles hve to e heked. Sine the sum of the interior ngles in ny tringle is onstnt numer (180 ), the third pir of orresponding ngles will utomtilly e equl, provided tht the first two pirs mth extly. Topi 10 edutive geometry 407
mesurement n geometry UNORRETE PGE PROOFS WorKe exmple 4 Find pir of similr tringles mong those shown. Give reson for your nswer. 2 m think Prove tht Δ is similr to ΔE. think 140 3 m WorKe exmple 5 6 m Exerise 10.3 Similr tringles InIVIuL PtHWys E WrIte 1 Write the informtion given. is prllel to E. Trnsversl forms two lternte ngles: nd E. 1 Given: Δ nd ΔE E is ommon. 2 Write wht is to e proved. 2 To prove: Δ ΔE 140 4 m 1 In eh tringle the lengths of two sides nd the inluded ngle re known, so the SS test n e pplied. Sine ll inluded ngles re equl (140 ), we need to the find rtios of orresponding sides, tking two tringles t time. 2 Only tringles nd hve orresponding sides in the sme rtio (nd inluded ngles re equl). Stte your onlusion, speifying the similrity test tht hs een used. 3 Write the proof. 3 Proof: = E (lternte ngles) = E (lternte ngles) = E (vertilly opposite ngles) Δ ΔE (equingulr, ) 3 m WrIte 140 5 m For tringles nd : 6 3 = 4 2 = 2 For tringles nd : 5 3 = 1.6, 3 2 = 1.5 For tringles nd : 5 6 = 0.83, 3 4 = 0.75 Tringle ~ tringle (SS) refletion How n you e ertin tht two fi gures re similr? PrtIse Questions: 1 10 onsolite Questions: 1 11, 15 mster Questions: 1 15 Individul pthwy intertivity int-4613 408 Mths Quest 10 + 10
mesurement n geometry FLueny 1 WE4 Find pir of similr tringles mong those shown in eh prt. Give reson for your nswer. UNORRETE PGE PROOFS i ii 5 5 3 4 i ii 4 2 20 5 20 2.5 i 2 ii 2 4 5 3 4 d i ii 40 60 50 60 e i ii 4 8 3 6 2 4 unerstning 2 Nme two similr tringles in eh of the following figures. Q P R d e iii iii iii iii iii 10 6 8 20 12 6 4.5 3 40 60 7 5 4 P R S Q T do 5278 do 5281 E E Topi 10 edutive geometry 409
mesurement N geometry UNORRETE PGE PROOFS RESONING 3 omplete this sttement: = = E. Find the vlue of the pronumerls. 4 Find the vlue of the pronumerl in the digrm t right. 5 The tringles shown t right re similr. Find the vlue of x nd y. 6 Stte why these two tringles shown t right re similr. Find the vlues of x nd y in the digrm. 7 wterslide is 4.2 m high nd hs support 2.4 m tll. If student rehes this support when she is 3.1 m down the slide, how long is the slide? 4.2 m 3.1 m 2.4 m 3 P Q P 20 9 4 45 2 4 4 3 2 1.5 8 R y x x 4 g 4 y E Q 45 R S T f x 1 6 410 Mths Quest 10 + 10
mesurement N geometry UNORRETE PGE PROOFS 8 storge tnk s shown in the digrm is mde of 4-m tll ylinder joined y 3-m tll one. If the dimeter of the ylinder is 5 m, wht is the rdius of the end of the one if 0.75 m hs een ut off the tip? 9 lulte the vlues of the pronumerls. 2 m (4x + 1) m 7 m 5 m y m 1.5 2x + 1 2 10 WE5 Prove tht Δ is similr to ΔE in eh of the following. E d E 11 Δ is right ngled tringle. line is drwn from to s shown so tht. Prove tht: Δ ~ Δ Δ ~ Δ 12 Explin why the test nnot e used to prove ongruene ut n e used to prove similrity. 13 Prove Pythgors theorem, 2 = 2 + 2, using similr tringles. E Prove the onverse of Pythgors theorem; tht is, if the squre on one side of tringle equls the sum of the squres on the other two sides, then the ngle etween these other two sides is right ngle. x 2 y 5 m 0.75 m 2.5 m E 4 m 3 m Topi 10 edutive geometry 411
mesurement n geometry ProLem solving 14 Prove tht ΔEFO ~ ΔGHO. UNORRETE PGE PROOFS E F O H 15 G Solve for x. x 2 4x 20 do 5282 3 E x 3 e 10.1 10.4 Qudrilterls Qudrilterls re four sided plne shpes whose interior ngles sum to 360. int-2786 Theorem 5 Theorem 5: The sum of the interior ngles in qudrilterl is 360. 412 Mths Quest 10 + 10 10edutiveGeometry.indd 412 8/20/14 6:31 PM
mesurement N geometry UNORRETE PGE PROOFS Given: To prove: onstrution: Proof: qudrilterl + + + = 360 rw line joining vertex to vertex. Lel the interior ngles of the tringles formed. e f + + = 180 (sum of interior ngles in tringle is 180 ) d + e + f = 180 (sum of interior ngles in tringle is 180 ) + + + d + e + f = 360 + + + = 360 Prllelogrms prllelogrm is qudrilterl with two pirs of prllel sides. Theorem 6 Theorem 6: Opposite ngles of prllelogrm re equl. Given: To prove: onstrution: 7 nd 7 = rw digonl from to. Proof: = (lternte ngles) = (lternte ngles) = + (y onstrution) = + (y onstrution) = onversely, if eh pir of opposite ngles of qudrilterl is equl then it is prllelogrm. d Topi 10 edutive geometry 413
mesurement N geometry Theorem 7 Theorem 7: Opposite sides of prllelogrm re equl. UNORRETE PGE PROOFS Given: To prove: onstrution: Proof: nd = rw digonl from to. = (lternte ngles) = (lternte ngles) is ommon to Δ nd Δ. Δ Δ (S) = onversely, if eh pir of opposite sides of qudrilterl is equl then it is prllelogrm. Theorem 8 Theorem 8: The digonls of prllelogrm iset eh other. Given: To prove: Proof: nd with digonls nd O = O nd O = O In ΔO nd ΔO, O = O (lternte ngles) O = O (lternte ngles) = (opposite sides of prllelogrm) ΔO ΔO (S) O = O (orresponding sides in ongruent tringles) nd O = O (orresponding sides in ongruent tringles) O 414 Mths Quest 10 + 10
mesurement N geometry Retngles retngle is prllelogrm with four right ngles. UNORRETE PGE PROOFS Theorem 9 Theorem 9: prllelogrm with right ngle is retngle. Given: Prllelogrm with = 90 To prove: = = = = 90 Proof: (properties of prllelogrm) + = 180 (o interior ngles) ut = 90 (given) = 90 Similrly, = = 90 = = = = 90 Theorem 10 Theorem 10: The digonls of retngle re equl. Given: Retngle with digonls nd To prove: = Proof: In Δ nd Δ, = (opposite sides equl in retngle) = (ommon) = = 90 (right ngles in retngle) Δ Δ (SS) = Rhomuses rhomus is prllelogrm with four equl sides. Theorem 11 Theorem 11: The digonls of rhomus re perpendiulr. O Topi 10 edutive geometry 415
mesurement N geometry UNORRETE PGE PROOFS Given: Rhomus with digonls nd To prove: Proof: In ΔO nd ΔO, O = O (property of prllelogrm) = (property of rhomus) O = O (ommon) ΔO ΔO (SSS) O = O ut O + O = 180 O = O = 90 Similrly, O = O = 90. Hene, (supplementry ngles) The midpoint theorem Now tht the properties of qudrilterls hve een explored, the midpoint theorem n e tkled. Theorem 12 Theorem 12: The intervl joining the midpoints of two sides of tringle is prllel to the third side nd hlf its length. Given: Δ in whih = nd E = E To prove: E nd E = 1 2 onstrution: rw line through prllel to. Extend E to F on the prllel line. Proof: In ΔE nd ΔEF, E = E E = EF E = EF ΔE ΔEF = F nd E = EF (E is the midpoint of, given) (vertilly opposite ngles) (lternte ngles) (S) (orresponding sides in ongruent tringles) E E F 416 Mths Quest 10 + 10
mesurement N geometry UNORRETE PGE PROOFS So, = = F. We hve F So F is prllelogrm. E lso, = F ut E = F E = 1 2 Therefore, E nd E = 1 2. (y onstrution) (opposite sides in prllelogrm) (sides in ongruent tringles) onversely, if line intervl is drwn prllel to side of tringle nd hlf the length of tht side, then the line intervl isets eh of the other two sides of the tringle. summry of the definitions nd properties of qudrilterls is shown in the tle. Shpe efinition Properties Trpezium Prllelogrm Rhomus Retngle Squre trpezium is qudrilterl with one pir of opposite sides prllel. prllelogrm is qudrilterl with oth pirs of opposite sides prllel. rhomus is prllelogrm with four equl sides. retngle is prllelogrm whose interior ngles re right ngles. squre is prllelogrm whose interior ngles re right ngles with four equl sides. One pir of opposite sides is prllel ut not equl in length. Opposite ngles re equl. Opposite sides re equl. igonls iset eh other. igonls iset eh other t right ngles. igonls iset the ngles t the vertex through whih they pss. igonls re equl. igonls iset eh other. ll ngles re right ngles. ll side lengths re equl. igonls re equl in length nd iset eh other t right ngles. igonls iset the vertex through whih they pss (45 ). Topi 10 edutive geometry 417
mesurement n geometry Reltionships etween qudrilterls The flowhrt elow shows the reltionships etween qudrilterls. UNORRETE PGE PROOFS refletion How do you know if qudrilterl is rhomus? do 5279 Kite Rhomus Qudrilterl Prllelogrm Squre Exerise 10.4 Qudrilterls InIVIuL PtHWys PrtIse Questions: 1 10 onsolite Questions: 1 14, 17 Individul pthwy intertivity int-4614 mster Questions: 1 20 Trpezium Retngle FLueny 1 Use the definitions of the five speil qudrilterls to deide if the following sttements re true or flse. squre is retngle. rhomus is prllelogrm. squre is rhomus. d rhomus is squre. e squre is trpezium. f prllelogrm is retngle. g trpezium is rhomus. h retngle is squre. 2 etermine the vlues of x nd y in eh of the following figures. (3x + 10) y x 3 m (2x 10) 4 m y 418 Mths Quest 10 + 10
mesurement N geometry 9x 11x d x UNORRETE PGE PROOFS UNERSTNING y 2x 3x y 3 rw three different trpeziums. Using your ruler, ompss nd protrtor, deide whih of the following properties re true in trpezium. Opposite sides re equl. ll sides re equl. Opposite ngles re equl. d ll ngles re equl. e igonls re equl in length. f igonls iset eh other. g igonls re perpendiulr. h igonls iset the ngles they pss through. 4 rw three different prllelogrms. Using your ruler nd protrtor to mesure, deide whih of the following properties re true in prllelogrm. Opposite sides re equl. ll sides re equl. Opposite ngles re equl. d ll ngles re equl. e igonls re equl in length. f igonls iset eh other. g igonls re perpendiulr. h igonls iset the ngles they pss through. 5 Nme four qudrilterls tht hve t lest one pir of opposite sides tht re prllel nd equl. 6 Nme qudrilterl tht hs equl digonls tht iset eh other nd iset the ngles they pss through. 7 Pool is plyed on retngulr tle. lls re hit with ue nd oune off the sides of the tle until they lnd in one of the holes or pokets. rw retngulr pool tle mesuring 5 m y 3 m on grph pper. Mrk the four holes, one in eh orner. ll strts t. It is hit so tht it trvels t 45 digonl ross the grid. When it hits the side of the tle, it ounes off t 45 digonl s well. How mny sides does the ll oune off efore it goes in hole? different size tle is 7 m y 2 m. How mny sides does ll oune off efore it goes in hole when hit from? Topi 10 edutive geometry 419
mesurement N geometry d omplete the following tle. UNORRETE PGE PROOFS Tle size 5 m 3 m 7 m 2 m 4 m 3 m 4 m 2 m 6 m 3 m 9 m 3 m 12 m 4 m Numer of sides hit e n you see pttern? How mny sides would ll oune off efore going in hole when hit from on n m n tle? f The ll is now hit from on 5 m 3 m pool tle. How mny different pths n ll tke when hit long 45 digonls? o these pths ll hit the sme numer of sides efore going in hole? oes the ll end up in the sme hole eh time? Justify your nswer. g The ll is now hit from long the pth shown. Wht type of tringles nd qudrilterls re formed y the pth of the ll with itself nd the sides of the tle? re ny of the tringles ongruent? h ll is hit from on 6 m y 3 m tle. Wht shpes re formed y the pth of the ll with itself nd the sides of the tle? Is there only one pth possile? i hllenge: ll is hit from long 45 digonls. The tle is m n. n you find formul to predit whih hole the ll will go in? j hllenge: Wht would hppen if the gme ws plyed on trpezoidl tle? RESONING 8 Prove tht the digonls of rhomus iset eh other. 9 is prllelogrm. X is the midpoint of nd Y is the midpoint of. Prove tht XY is lso prllelogrm. X Y 420 Mths Quest 10 + 10
mesurement N geometry UNORRETE PGE PROOFS 10 is prllelogrm. P, Q, R nd S re ll midpoints of their respetive sides of. Prove ΔPS ΔRQ. Prove ΔSR ΔPQ. Hene, prove tht PQRS is lso prllelogrm. 11 nd re dimeters of irle with entre O. Prove tht is retngle. 12 The digonls of prllelogrm meet t right ngles. Prove tht the prllelogrm is rhomus. 13 Two ongruent right ngled tringles re rrnged s shown. Show tht PQRS is prllelogrm. 14 Two irles, entred t M nd N, hve equl rdii nd interset t P nd Q. Prove tht PNQM is rhomus. 15 Give resons why squre is rhomus, ut rhomus is not neessrily squre. 16 is trpezium. (x + 4) (x 4) y Wht ft do you know out trpezium? Find the vlues of x nd y. 17 is kite where = 8 m, E = 5 m nd E = 9 m. Find the ext vlues of: i x ii y. Find ngle nd hene ngle. (x + 15) P S E M S P Q R O Q R P N Q x y Topi 10 edutive geometry 421
mesurement n geometry ProLem solving E E is regulr pentgon whose 2 side lengths re 2 m. Eh digonl m is x m long. Wht kind of shpe is EF nd wht x m is the length of F? Wht kind of shpe is? If E is 40º, find the vlue of, giving resons for your findings. F d Whih tringle is similr to E? e Explin why F = (x 2) m. f Show tht x2 2x 4 = 0. g Solve the eqution x2 2x 4 = 0, giving your nswer s n ext vlue. 19 is lled yli qudrilterl euse it is insried inside irle. UNORRETE PGE PROOFS 18 422 (3x 35) do-5283 (2x + 35) E hrteristi of yli qudrilterl is tht the opposite ngles re supplementry. Find the vlue of x. Mths Quest 10 + 10 10edutiveGeometry.indd 422 8/20/14 6:31 PM
mesurement n geometry 10.5 Polygons Polygons re losed shpes tht hve three or more stright sides. UNORRETE PGE PROOFS Irregulr polygon Not polygon Not polygon Regulr polygons re polygons with sides of the sme length nd interior ngles of the sme size, like the pentgon shown in the entre of the photo ove. onvex polygons re polygons with no interior reflex ngles. onve polygons re polygons with t lest one reflex interior ngle. For exmple, the pentgon shown ove is onve polygon s well s regulr polygon. Interior ngles of polygon The interior ngles of polygon re the ngles inside the polygon t eh vertex. The sum of the interior ngles of polygon is given y the formul: WorKe exmple 6 ngle sum = 180 (n 2) where n = the numer of sides of the polygon lulte the vlue of the pronumerls in the figure elow. think 1 ngles nd 110 form stright line nd so re supplementry (dd to 180 ). 2 The interior ngles of tringle sum to 180. 3 Sustitute 70 for nd solve for. 80 110 150 WrIte + 110 = 180 + 110 110 = 180 110 = 70 + + 80 = 180 + 70 + 80 = 180 + 150 = 180 = 30 Topi 10 edutive geometry 423
mesurement n geometry UNORRETE PGE PROOFS refletion How re the ngles ssoited with polygons relted to eh other nd the polygon? int-0818 int-0819 Exterior ngles of polygon The exterior ngles of polygon re formed y the side of the polygon nd n extension of its djent side. For exmple, x, y nd z re externl ngles for the polygon (tringle) elow. y x z The exterior ngle nd interior ngle t tht vertex re supplementry (dd to 180 ). For exmple, x + = 180. Exterior ngles of polygons n e mesured in lokwise or ntilokwise diretion. In regulr polygon, the size of the exterior ngle n e found y dividing 360 y the numer of sides. Exterior ngle = 360 n The sum of the exterior ngles of polygon equls 360. The exterior ngle of tringle is equl to the sum of the opposite interior ngles. Exerise 10.5 Polygons InIVIuL PtHWys PrtIse Questions: 1 7, 9 onsolite Questions: 1 9, 12 q t r s mster Questions: 1 12 FLueny 1 How re the internl nd externl ngles of polygon relted to the numer of sides in polygon? 2 WE6 lulte the vlues of the pronumerls in the digrms elow. m 120 15 (t 10) 10 Individul pthwy intertivity int-4615 d 160 70 5x 424 Mths Quest 10 + 10
mesurement N geometry 3 For the five tringles elow, evlute the pronumerls nd determine the size of the interior ngles. y 15 160 UNORRETE PGE PROOFS e 55 n n d 18 (3t + 10) 20 92 (2t 2) t 4x 105 4 For the five qudrilterls elow: i lel the qudrilterls s regulr or irregulr ii determine the vlue of the pronumerl for eh shpe. e x 120 80 65 4t t 2m 3m d p 2y 70 20 y x 60 l Topi 10 edutive geometry 425
mesurement N geometry UNERSTNING 5 The photogrph elow shows house uilt on the side of hill. Use your knowledge of ngles to lulte the vlues of the pronumerls. UNORRETE PGE PROOFS x 133º 6 lulte the vlues of the four interior ngles of the front fe of the uilding in the photogrph elow. x + 15 y w x z 105º 426 Mths Quest 10 + 10
mesurement N geometry 7 lulte the vlues of the pronumerls for the irregulr polygons elow. f UNORRETE PGE PROOFS 350 m n d 120 o e 8 lulte the size of the exterior ngle of regulr hexgon (6 sides). RESONING 9 digonl of polygon joins two verties. lulte the numer of digonls in regulr polygon with: i 4 sides ii 5 sides iii 6 sides iv 7 sides. Write formul tht reltes the numer of digonls for n n sided polygon. 10 The externl ngle of polygon n e lulted using the formul: exterior ngle = 360 n Use the reltionship etween internl nd externl ngles of polygon to write formul for the internl ngle of regulr polygon. PROLEM SOLVING 11 J E F 8 m piee of string is fixed t nd H s shown. The string is tight nd fixed to the surfe of the uoid. Lote the ext position of J on the edge. 4 m H 3 m G Topi 10 edutive geometry 427
mesurement n geometry 12 EFGH is n otgon. UNORRETE PGE PROOFS do-5284 H 3x (3x 10) 3x Wht is the sum of the interior ngles of n otgon? Find the vlue of x. 2x (3x 10) G (4x + 20) F (3x + 5) 4x E 428 Mths Quest 10 + 10
mesurement n geometry UNORRETE PGE PROOFS ONLINE ONLY 10.6 Review www.jplus.om.u The Mths Quest Review is ville in ustomisle formt for students to demonstrte their knowledge of this topi. The Review ontins: Flueny questions llowing students to demonstrte the skills they hve developed to effiiently nswer questions using the most pproprite methods Prolem solving questions llowing students to demonstrte their ility to mke smrt hoies, to model nd investigte prolems, nd to ommunite solutions effetively. summry of the key points overed nd onept mp summry of this topi re ville s digitl douments. Review questions ownlod the Review questions doument from the links found in your eookplus. Lnguge int-2853 int-2854 iset onve ongrueny onvex orresponding ngles orresponding sides digonl exterior ngle interior ngle prllelogrm polygon qudrilterl retngle reflex rhomus similrity squre trpezium int-3597 Link to ssesson for questions to test your rediness For lerning, your progress s you lern nd your levels of hievement. ssesson provides sets of questions for every topi in your ourse, s well s giving instnt feedk nd worked solutions to help improve your mthemtil skills. the story of mthemtis is n exlusive Jrnd video series tht explores the history of mthemtis nd how it helped shpe the world we live in tody. <Text to ome> www.ssesson.om.u Topi 10 edutive geometry 429 10edutiveGeometry.indd 429 8/20/14 6:31 PM
mesurement n <InVestIgtIon> InVestIgtIon For geometry rih tsk or <mesurement n geometry> For PuZZLe UNORRETE PGE PROOFS rih tsk Enlrgement tivity The geometril properties shred y shpe nd its imge under enlrgement n e listed s: lines re enlrged s lines sides re enlrged to orresponding sides y the sme ftor mthing ngles on the two shpes re equl. In this tivity, we will strt with smll rtoon hrter, nd then low it up to lmost life size. Equipment: ruler, penil, rtoon print, uther s pper or some other lrge piee of pper. 1 o some reserh on the internet nd selet rtoon hrter or ny hrter of your hoie. 2 rw grid of 2-m squres over the smll rtoon hrter. Exmple: The smll sper is 9 squres wide nd 7 squres tll. 430 Mths Quest 10 + 10 10edutiveGeometry.indd 430 8/20/14 6:31 PM
mesurement n geometry UNORRETE PGE PROOFS 3 Lel the grids with letters ross the top row nd numers down the fi rst olumn, s shown in the exmple. 4 Get lrge piee of pper nd drw the sme numer of squres. You will hve to work out the rtio of similitude (e.g. 2 m : 8 m). 5 If your smll rtoon hrter strethes from one side of the smll pper (the pper the imge is printed on) to the other, your lrge sper must streth from one side of the ig pper to the other. Your lrge grid squres my hve to e 8 m y 8 m or lrger, depending on the pper size. 6 rw this enlrged grid on your lrge pper. Use metre ruler or some other long stright edged tool. e sure to keep ll of your squres the sme size. t this point, you re redy to RW. Rememer, you do NOT hve to e n rtist to produe n impressive enlrgement. ll you do is drw EXTLY wht you see in eh smll ell into its orresponding lrge ell. For exmple, in ell 3 of the sper enlrgement, you see the tip of his fi nger, so drw this in the ig grid. If you tke your time nd re very reful, you will produe n extremely impressive enlrgement. Wht you hve used is lled RTIO OF SIMILITUE. This rtio ontrols how lrge the new piture will e. 2 : 5 rtio will give you smller enlrgement thn 2 : 7 rtio, euse for every two units on the originl you re generting only 5 units of enlrgement insted of 7. If sper s rtio is 1 : 4, it produes fi gure tht hs liner mesure tht is four times igger. ig sper s overll re, however, will e 16 times lrger thn smll sper s. This is euse re is found y tking length times width. The length is 4 times longer nd the width is 4 times longer. Thus the re is 4 4 = 16 time s lrger thn the originl sper. His overll VoLume will e 4 4 4 or 64 times lrger! This mens tht ig sper will weigh 64 times more thn smll sper. Topi 10 edutive geometry 431
<InVestIgtIon> mesurement n For geometry rih tsk or <mesurement n geometry> For PuZZLe oe PuZZLe UNORRETE PGE PROOFS Why ws the rheologist upset? The tringles in eh pir re ongruent. Find the unknown in eh to solve the puzzle. 9 26 12 x 4x U 25 36 8 16 O 36 20 25 2x 20 12 2x 25 25 15 15 8 9 S x H 24 J W 20 35 30 130 9 130 3 8 9 4x 2x 5 5 3 7 8x 7 112 9x 60 5x I 12 30 N 12 60 R 8 8 2x 4 30 81 10 4 60 30 8 14 13 35 6 30 5 4 13 14 9 2 90 14 9 13 432 Mths Quest 10 + 10
mesurement n geometry tivities UNORRETE PGE PROOFS 10.1 overview Video The story of mthemtis (eles-1849) 10.2 ngles, tringles nd ongruene Intertivity IP intertivity 10.2 (int-4612) ngles, tringles nd ongruene igitl dos SkillSHEET (do 5276): Nming ngles, lines nd fi gure s SkillSHEET (do 5277): orresponding sides nd ngles of ongruent tringles SkillSHEET (do 5280): ngles nd prllel lines 10.3 similr tringles Intertivity IP intertivity 10.3 (int-4613) Similr tringles igitl dos SkillSHEET (do 5278): Writing similrity sttements SkillSHEET (do 5281): lulting unknown side lengths in pir of similr tringles WorkSHEET 10.1 (do 5282): edutive geometry I to ess eookplus tivities, log on to 10.4 Qudrilterls Intertivities Qudrilterl defi nitions (int 2786) IP intertivity 10.4 (int-4614) Qudrilterls igitl dos SkillSHEET (do 5279): Identifying qudrilterls WorkSHEET 10.2 (do-5283): edutive geometry II 10.5 Polygons Intertivities ngle sum of polygon (int-0818) Exterior ngles of polygon (int-0819) IP intertivity 10.5 (int-4615) Polygons igitl do WorkSHEET 10.3 (do-5284): edutive geometry III 10.6 review Intertivities Word serh (int 2853) rossword (int 2854) Sudoku (int-3597) igitl dos Topi summry (do-13761) onept mp (do-13762) www.jplus.om.u Topi 10 edutive geometry 433
mesurement N geometry nswers topi 10 edutive geometry Exerise 10.2 ngles, tringles nd ongruene 1 = 56 = 30 = 60 d d = 120 e e = 68 2 I nd III, SS I nd II, S II nd III, RHS d I nd II, SSS 3 x = 6, y = 60 x = 80, y = 50 x = 32, y = 47 d x = 45, y = 45 4 x = 3 m x = 85 x = 80, y = 30, z = 70 d x = 30, y = 7 m e x = 40, y = 50, z = 50, m = 90, n = 90 5 Use SS Use SS. Use S. d Use S. e Use SSS. 6, 7 x = 110, y = 110, z = 4 m, w = 7 m x = 70 x = 30, y = 65 8 The third sides re not neessrily the sme. 9 orresponding sides re not the sme. 10 Use SSS. 11, 12 hek with your teher. 13 hek with your teher. 14 x = 20, y = 10 nd z = 40 UNORRETE PGE PROOFS Exerise 10.3 Similr tringles 1 i nd iii, RHS i nd ii, SS i nd iii, SSS d i nd iii, e i nd ii, SSS 2 Tringles PQR nd Tringles nd Tringles PQR nd TSR d Tringles nd E e Tringles nd E 3 = E = f = 9, g = 8 E 4 x = 4 5 x = 20, y = 2 1 4 6 x = 3, y = 4 7 The slide is 7.23 m long. 8 Rdius = 0.625 m 9 x = 1, y = 7!2 x = 2 1, y = 3.9 2 10 hek with your teher. 11 = (ommon ngle) = = 90 Δ ~ Δ () = (ommon ngle) = = 90 Δ ~ Δ () 12 ongruent tringles must e identil, tht is, the ngles must e equl nd the side lengths must e equl. Therefore, it is not enough just to prove tht the ngles re equl. 13 hek with your teher. 14 FEO = OGH (lternte ngles equl s EF HG) EFO = OHG (lternte ngles equl s EF HG) EOF = HOG (vertilly opposite ngles equl) ΔEFO ΔGHO (equingulr) 15 x = 6 or 11 hllenge 10.1 3 Exerise 10.4 Qudrilterls 1 True True True d Flse e Flse f Flse g Flse h Flse 2 x = 36, y = 62 x = 5 m, y = 90 x = 10, y = 70 d x = 40, y = 60 3 None re true, unless the trpezium is regulr trpezium, then e is true. 4,, f 5 Prllelogrm, rhomus, retngle, squre 6 Squre 7 6 sides 7 sides d Tle size Numer of sides hit 5 m 3 m 6 7 m 2 m 7 4 m 3 m 5 4 m 2 m 1 6 m 3 m 1 9 m 3 m 2 12 m 4 m 2 e If the rtio of the sides is written in simplest form then the pttern is m + n 2. f There re two routes for the ll when hit from. Either 2 or 3 sides re hit. The ll does not end up in the sme hole eh time. suitle justifition would e digrm student to drw. g Isoseles tringles nd prllelogrms. The tringles re ongruent. h The shpes formed re prllelogrms. There is only one possile pth lthough the ll ould e hit in either of two diretions initilly. i Given m : n is the rtio length to width in simplest form. When m is even nd n is odd the destintion poket will e the upper left. When m nd n re oth odd, the destintion poket will e the upper right. When m is odd nd n is even the destintion poket will e the lower right. j Students to investigte. 8 hek with your teher. 9 X Y euse is prllelogrm X = Y (given) XY is prllelogrm sine opposite sides re equl nd prllel. 10 Use SS. Use SS. Opposite sides re equl. 11 = (dimeters of the sme irle re equl) O = O nd O = O (rdii of the sme irle re equl) is retngle. (igonls re equl nd iset eh other.) 12 hek with your teher. 13 PS = QR (orresponding sides in ongruent tringles re equl) PS QR (lternte ngles re equl) PQRS is prllelogrm sine one pir of opposite sides re prllel nd equl. 14 MP = MQ (rdii of sme irle) PN = QN (rdii of sme irle) nd irles hve equl rdii. ll sides re equl. PNQM is rhomus. 434 Mths Quest 10 + 10
mesurement N geometry UNORRETE PGE PROOFS 15 hek with your teher. 16 One pir of opposite sides re prllel. x = 90, y = 75 17 i x = "41 ii y =!97 = = 117 23 18 Rhomus, 2 m Trpezium 40 d Tringle F e hek with your teher. f hek with your teher. g x = (1 +!5) m 19 70 hllenge 10.2 x =!10 m Exerise 10.5 Polygons 1 The sum of the interior ngles is sed on the numer of sides of the polygon. The size of the exterior ngle n e found y dividing 360 y the numer of sides. 2 m = 60 = 45, = 45 t = 35 d x = 10 3 y = 35 t = 5 n = 81 d x = 15 e t = 30 4 i Irregulr ii x = 95 i Irregulr ii p = 135 i Irregulr ii t = 36 d i Irregulr ii y = 70 e i Irregulr ii p = 36 5 w = 75, x = 105, y = 94, z = 133 6 82.5, 82.5, 97.5, 97.5 7 = 120, = 120, = 60, d = 60, e = 120, f = 240 m = 10, n = 270, o = 50 8 60 9 i 2 ii 5 iii 9 iv 14 Numer of digonls = 1 n(n 3) 2 10 Internl ngle = 180 360 n 11 J is 24 m from. 7 12 1080 43 Investigtion Rih tsk hek with your teher. ode puzzle His jo ws in ruins. Topi 10 edutive geometry 435
mesurement n geometry IT tivity UNORRETE PGE PROOFS Getting the udget in order serhlight I: Pro-0087 senrio udgets re of gret importne to everyone. From the federl udget down to the lol ounil udget, they ll impt on our lives. On smller sle, ut of no less importne, is the fmily udget. s fi nnil plnner, you hve een sked to prepre udget for the Thompson fmily. The Thompson fmily lives in mjor ity in ustrli nd omprises Mr nd Mrs Thompson nd their two teenge hildren. They wnt to go on holidy nd they pln to sve for yer. Tking their ongoing expenses into ount, they wnt to see if nd how they n fford this, following your reommendtions. You re to form fi nnil plnning group of three people. Eh group memer is to produe udget pln tht ddresses the following holidy nd ongoing expenses. Holidy: They wnt to e le to spend $5000 on holidy within ustrli. Where n they go nd for how long, tking fl ights, ommodtion, mels nd sightseeing expenses into ount? Phone plns: Their two hildren hve phone plns tht eh ost $29 per month, urrently pid y their prents. However, to help fund the holidy, their prents need them to py k the $29 per month, with either simple interest or ompound interest hrged. Wht is the est option for the hildren? How muh do they need to work t their $9.75 per hour prt time jos to over the ost? Rent: The Thompson fmily lives in rentl house within 50 km from mjor ity. How muh rent would they e pying? Txes Utilities (eletriity, wter, gs, phone) Food Other expenses Within the Thompson fmily, Mr Thompson tkes re of the household duties nd Mrs Thompson works in slried position, erning $85 000 per nnum. Their hildren hve prt time jos to repy the ost of their phone plns nd to try nd sve for new lothes. tsk You will produe n orl presenttion nd written udget pln with reommendtions for the Thompsons fmily holidy. The udget pln will inlude the preferred method of interest hrged to the hildren for the ost of their phone pln nd the hours per week the hildren need to work t their prt time jos to mke this repyment. Your udget pln should lso inlude vlue for money rentls loted within 50 km of 436 Mths Quest 10 + 10
mesurement n geometry UNORRETE PGE PROOFS mjor ity, whih will help the fmily ssess how muh they should e spending on rent. You lso need to reommend pkge holidy tht inludes fl ights nd ommodtion. Your presenttion should explin why you hve mde eh of your reommendtions nd give evidene to k up your deisions, tking ll the expenses into ount. Proess Open the ProjetsPLUS pplition for this hpter in your eookplus. Wth the introdutory video lesson, lik the Strt Projet utton nd then set up your projet group. You n omplete this projet individully or invite other memers of your lss to form group. Sve your settings nd the projet will e lunhed. Nvigte to your Reserh Forum. Here you will fi nd series of topis tht will help you omplete your tsk. Selet the expenses you re reserhing or dd new expenses you wish to inlude. Reserh. Mke notes of importnt informtion nd ides tht you disovered during your reserh. Enter your fi ndings s rtiles under your topis in the Reserh forum. You should eh fi nd t lest three soures of informtion (inluding off line resoures suh s ooks nd newsppers). You n view nd omment on other group memers rtiles nd rte the informtion they suggeste softwre ProjetsPLUS Mirosoft Word PowerPoint, Prezi, Keynote or other presenttion softwre Mirosoft Exel S lultor (optionl) hve entered. When your reserh is omplete, print your Reserh Report to hnd in to your teher. Visit your Medi entre nd downlod the udget templte nd PowerPoint smple to help you prepre your presenttion. Your Medi entre lso inludes imges to help liven up your presenttion. Use the udget templte to give ler overview of ll the expenses tken into ount nd to give n overll summry of the whole fmily udget. Mke sure you rememer to ddress ll the expenses tht the Thompson fmily hs requested you tke into ount. Use the PowerPoint templte to develop your presenttion. Rememer tht you re mking reommendtions tht you elieve re est for the Thompson fmily. Mke sure you over ll the detils they tht hve requested, nd tht your presenttion will gr their ttention. Topi It 10 tivity edutive Projetsplus geometry 437