ganit (6) - रेखाएं और कोण

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File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-6\Chap-6.PM65

v;k;6

js[kk,vkSj dks.k

6.1 Hkwfedk

v;k; 5 esa] vki iksaiM+h dk ekWMy cukuk pkgrsgSaA lksfp,] vki bls oSQlscuk,xsA oqQN cklksa dks vki ijLij lekarj j[ksaxsvkSj oqQN dksfrjNk j[ksaxsA tc ,d vkfoZQVsDV (architect),d cgqryh; Hkou osQ fy, ,d js[kkfp=k[khaprk gS] rks mls fofHkUu dks.kksaij izfrPNsnh vkSj lekarj js[kk,[khapuh iM+rh gSaA D;k

vki lksprsgSafd og js[kkvksavkSj dks.kksaosQ Kku ds fcuk bl Hkou dh :ijs[kk [khapldrk gS\

foKku esa] vki izdk'k osQ xq.kksa dk fdj.k vkjs[k (ray diagrams)[khap dj v;;udjrsgSaA mnkgj.kkFkZ] izdk'k osQ viorZu (refraction)xq.k dk v;;u djusosQ fy,] tc
http://schools.aglasem.com/

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js[kk,vkSj dks.k 109


izdk'k dh fdj.ksa ,d ek;e (medium)ls nwljsek;e esaizos'k djrh gSa] vki izfrPNsnhjs[kkvksa vkSj lekarj js[kkvksaosQ xq.kksadk iz;ksx djrsgSaA tc ,d fiaM ij nks;k vf/dcy dk;Z dj jgs gksa] rks vki bu cyksadk ml fiaM ij ifj.kkeh cy Kkr djus osQ fy,],d ,slk vkjs[k [khaprs gSa ftlesa cyksa dks fn"V js[kk[kaMksa (directed line segments)}kjkfu:fir fd;k tkrk gSA ml le;] vkidksmu dks.kksaosQ chp laca/ tkuus dh vko';drkgksxh ftudh fdj.ksa (vFkok js[kk[kaM) ijLij lekarj ;k izfrPNsnh gksaxhA ,d ehukj dhpkbZKkr djus vFkok fdlh tgkt dh ,d izdk'k iqat (light house)ls nwjh Kkr djusosQ fy,] gesa {kSfrt vkSj n`f"V js[kk (line of sight)osQ chp cus dks.k dh tkudkjh dhvko';drk gksxhA izpqj ek=kk esa,slsmnkgj.k fn, tk ldrs gSa tgk js[kkvksavkSj dks.kksa dk

iz;ksx fd;k tkrk gSAT;kfefr osQ vkusokysv;k;ksa esa] vki js[kkvksa vkSj dks.kksa osQ bu xq.kksadk vU; mi;ksxh xq.kksa dks fuxfer (fudkyus) djus esa iz;ksx djsaxsA

vkb, igys ge fiNyh d{kkvksa esa js[kkvksa vkSj dks.kksals lacaf/r i

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(i)U;wu dks.k: 0

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'kh"kZ gSA fdj.k BA vkSj fdj.k BCosHkqtk,gSatksmHk;fu"B ughagSaA blosQ vfrfjDr]tc nks dks.k vklUu dks.k gksrsgSa] rks mudk ;ksx ml dks.k osQ cjkcj gksrk gStks budh

mu Hkqtkvksa ls curk gS] tks mHk;fu"B ugha gSaA vr% ge fy[k ldrs gSa fd ABC = ABD + DBCgSA

;ku nhft, fd ABC vkSj ABD vklUu

dks.k ugha gSaA D;ksa\ bldk dkj.k ;g gS fdvmHk;fu"B Hkqtk,(vFkkZr~os Hkqtk, tksmHk;fu"B

ughagSa)BD vkSjBCmHk;fu"B HkqtkBAosQ ,dgh vksj fLFkr gSA

;fn vkfr 6-2 esa] vmHk;fu"B Hkqtk, BAvkSjBC,d js[kk cuk,] rks;g vkfr 6-3 tSlk

yxsxkA bl fLFkfr esa] ABD vkSj DBCdks.kksa

dk ,d jSf[kd ;qXe(linear pair of angles)cukrsgSaA

vki'kh"k kZfH ke q[k d ks.kksa(vertically opposite

angles)dks Hkh ;kn dj ldrsgSa] tks nks js[kkvksa]eku yhft,] AB vkSjCDdksijLij fcanqOij

izfrPNsn djusij cursgSa (nsf[k, vkfr 6-4)A;gk'kh"kkZfHkeq[k dks.kksa osQ nks;qXe gSaA buesa ls ,d;qXe AOD vkSj BOCdk gSA D;k vki nwljk

;qXe Kkr dj ldrs gSa\

6.3izfrPNsnh js[kk, vkSj vizfrPNsnh js[kk,

,d dkxt ij nks fHkUu js[kk,PQ vkSjRS[khafp,Avki ns[ksaxs fd vki bu js[kkvksa dksnksizdkj ls [khap ldrsgSa] tSlk fd vkfr 6.5 (i) vkSj vkfr6.5 (ii)esa n'kkZ;k x;kgSA

vkfr6.4 :'kh"kkZfHkeq[k dks.k

vkfr6.3 :dks.kksa dk jSf[kd ;qXe

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(i)izfrPNsnh js[kk, (ii) vizfrPNsnh (lekarj) js[kk,

vkfr6.5 :nksjs[kk,[khapus osQ fofHkUu izdkj

js[kk dh bl vo/kj.kk dks Hkh ;kn dhft, fd og nksuksa fn'kkvksa esavfuf'pr :i

ls foLr`r gksrh gSA js[kk,PQ vkSj RS vkfr 6.5 (i)esa izfrPNsnh js[kk, gSa vkSjvkfr6.5 (ii)esa ;slekarj js[kk,gSaA ;ku nhft, fd bu nksuksalekarj js[kkvksa osQ fofHkUufcanqvksaij muosQ mHk;fu"B yEcksadh yackb;k leku jgsaxhA ;g leku yackbZnksuksa lekarjjs[kkvksa osQ chp dh nwjh dgykrh gSA

6.4dks.kk saosQ ;qXe

vuqPNsn 6-2 esa] vki dks.kksa osQ oqQN ;qXeksa tSlsiwjd dks.k] laiwjd dks.k] vklUu dks.k] dks.kksa dkjSf[kd ;qXe] bR;kfn dh ifjHkk"kkvksaosQ ckjs esai

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vfHkx`ghr6.1 :;fn ,d fdj.k ,d js[kk ij [kM+h gks] rksbl izdkj cus nksuk sa vklUudks.kksa dk ;ksx 180gksrk gSA

;kn dhft, fd tc nks vklUu dks.kksadk ;ksx 180gks] rksosdks.kksadk ,djSf[kd;qXecukrsgSaA

vfHkxghr 6-1 esa;g fn;k gS fd ,d fdj.k ,d js[kk ij [kM+h gks*A bl fn, gq, ls]geusfu"d"kZ fudkyk fd bl izdkj cusnksuksa vklUu dks.kksadk ;ksx 180gksrk gSA D;k gevfHkxghr 6-1 dks,d foijhr izdkj ls fy[k ldrsgSa\ vFkkZr~vfHkx`ghr 6-1 osQ fu"d"kZdksfn;k gqvk ekusa vkSj mlosQ fn, gq, dksfu"d"kZ ekusaA rc gesa ;g izkIr gksxk%

(A);fn nksvklUu dks.kksadk ;ksx 180gS] rks ,d fdj.k ,d js[kk ij [kM+h gksrhgS(vFkkZr~vmHk;fu"B Hkqtk,,d gh js[kk esagSa)A

vc vki ns[krsgSafd vfHkx`ghr6.1 vkSj dFku(A),d nwljsosQ foijhr gSaA ge buesalsizR;sd dks nwljsdkfoykse(converse)dgrsgSaA ge ;g ughatkursfd dFku (A)lR; gS;k ughaA vkb, bldh tkp djsaaA fofHkUu ekiksa osQ] vkfr 6-7 esa n'kkZ, vuqlkj]vklUu dks.k [khafp,A izR;sd fLFkfr esa] vmHk;fu"B Hkqtkvksaesa ls ,d Hkqtk osQ vuqfn'k,d iVjh (ruler)jf[k,A D;k nwljh Hkqtk Hkh bl iVjh osQ vuqfn'k fLFkr gS\

vkfr6.7 :fofHkUu ekiksaosQ vklUu dks.k

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vki ik,xs fd osQoy vkfr 6.7 (iii)esagh nksuksa vmHk;fu"B Hkqtk, iVjh osQ vuqfn'kgSa] vFkkZr~ A, O vkSjB,d gh js[kk ij fLFkr gSavkSj fdj.k OCbl js[kk ij [kM+h gSAlkFk gh] ;g Hkh nsf[k, fd AOC + COB = 125 + 55 = 180gSAblls vki fu"d"kZfudky ldrsgSafd dFku(A)lR; gSAvr%] vki bls,d vfHkx`ghr osQ :i esa fuEuizdkj fy[k ldrs gSa%

vfHkx`ghr6.2 :;fn nksvklUu dks.kksa dk ;ksx 180gS] rksmudh vmHk;fu"B Hkqtk,,djs[kk cukrh gSaA

Li"V dkj.kksa ls] mijksDr nksuksa vfHkx`ghrksadksfeyk djjSf[kd ;qXe vfHkx`ghr(Linear Pair Axiom)

dgrsgSaAvkb, vc ml fLFkfr dh tkp djsa tc nksjs[kk, izfrPNsn djrh gSaA

fiNyh d{kkvksa ls vkidks ;kn gksxk fd ;fn nksjs[kk, ijLij izfrPNsn djsa] rks'kh"kZkfHkeq[k dks.k cjkcj gksrs gSaA vkb, vc bl ifj.kke dks fl djsaA ,d miifk

(proof)esafufgr vo;oksa osQ fy,] ifjf'k"V 1 dksnsf[k, vkSj uhpsnh gqbZmiifk dksi

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blh izdkj] fl fd;k tk ldrk gSfdAOD = BOCgSA nvkb, vc jSf[kd ;qXe vfHkx`ghr vkSj izes; 6-1 ij vk/kfjr oqQN mnkgj.k gy djsaA

mnkgj.k1 : vkfr 6-9 esa] js[kk,PQ vkSjRSijLijfcanq Oij izfrPNsn djrh gSaA ;fn POR : ROQ

= 5 : 7gS] rks lHkh dks.k Kkr dhft,A

gy: POR + ROQ = 180(jSf[kd ;qXe osQ dks.k)

ijUrq] POR : ROQ = 5 : 7 (fn;k gS)

vr%] POR = 51 180 = 75

blh izdkj] ROQ =7

1 180 = 105

vc POS = ROQ = 105 ('kh"kkZfHkeq[k dks.k)

vkSj SOQ = POR = 75 ('kh"kkZfHkeq[k dks.k)

mnkgj.k2 :vkfr 6-10 esa] fdj.k OSjs[kk POQij [kM+h gSA fdj.k OR vkSj OTe'k% POS vkSj SOQosQ lef}Hkktd gSaA ;fn POS =xgS] rks ROTKkr

dhft,Agy:fdj.kOSjs[kkPOQij [kM+h gSA

vr%, POS + SOQ = 180

ijUrq] POS =x

vr%] x+ SOQ = 180

blfy,] SOQ = 180 x

vc fdj.k OR, POSdks lef}Hkkftr djrh gSA

blfy,] ROS = 12 POS

=1

2x=

2

x

vko`Qfr6.9

vko`Qfr6.10

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blh izdkj] SOT =1

2 SOQ

=1

2 (180 x)

= 9 02

x -

vc] ROT = ROS + SOT

= 9 0 2 2

x x+

= 90

mnkgj.k3 :vkfr 6-11 esa]OP, OQ, OR vkSjOSpkj fdj.ksagSaA fl dhft, fd POQ + QOR + SOR + POS = 360gSA

gy:vkfr6.11esa] vkidks fdj.kksaOP, OQ, ORvkSjOSesa lsfdlh ,d dksihNs ,d fcanq rd

c

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vc](1) vkSj(3)dks tksM+us ij] vkidks izkIr gksxk% TOP + POQ + TOS + SOR + QOR = 360 (4)

ijUrq TOP + TOS = POSgSA

vr%](4)fuEu gks tkrh gS % POQ + QOR + SOR + POS = 360

iz'ukoyh6.1

1. vkfr6.13esa,js[kk,AB vkSjCDfcanqOij izfrPNsn

djrh gSaA ;fn AOC + BOE = 70gS vkSj BOD = 40gS] rksBOEvkSj izfrorhZCOEKkr

dhft,A

2. vkfr 6-14 esa] js[kk, XY vkSj MNfcanq OijizfrPNsn djrh gSaA;fn POY = 90 vkSja: b= 2 : 3gS,rkscKkr dhft,A

3. vkfr6.15esa];fn PQR = PRQgS] rksfldhft, fd PQS = PRTgSA

vko`Qfr 6.13

vko`Qfr 6.14

vko`Qfr 6.15

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4. vkfr6.16esa,;fnx+y= w+ zgS,rks fl dhft,fd AOB,d js[kk gSA

5. vkfr 6.17esa, POQ,d js[kk gSA fdj.k ORjs[kkPQij yEc gSA fdj.kksaOP vkSj ORosQ chp esa OS

,d vU; fdj.k gSA fl dhft,%

ROS =1

2( QOS POS)

6. ;g fn;k gS fdXYZ = 64gS vkSj XYdksfcanqPrd c

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1, 2, 7 vkSj 8ck% dks.k(exterior angles)dgykrs gSaA 3, 4, 5 vkSj 6 var% dks.k(interior angles)dgykrs gSaA

;kn dhft, fd fiNyh d{kkvksaesa] vkius oqQN dks.kksa osQ ;qXeksa dk ukekadu fd;k Fkk]tks,d fr;Zd js[kk }kjk nks js[kkvksa dks izfrPNsn djusls curs gSaA ;s ;qXe fuEu gSa%

(a)laxr dks.k (Corresponding angles) :

(i) 1 vkSj 5 (ii) 2 vkSj 6

(iii) 4 vkSj 8 (iv) 3 vkSj 7

(b),dkarj var% dks.k (Alternate interior angles) :


(c),dkarj ck% dks.k (Alternate exterior angles) :


(d)fr;Zd js[kk osQ ,d gh vksj osQ var% dks.k%


fr;Zd js[kk osQ ,d gh vksj osQ var% dks.kksa dksekxr var%dks.k(consecutive interior angles);klacaf/r dks.k(alliedangles);klg&var% dks.k(co-interior angles)Hkh dgk tkrk gSA lkFk gh] vusd ckj ge ,dkarj var% dks.kksa osQfy, osQoy 'kCnksa,dkarj dks.kksadk iz;ksx djrsgSaA

vkb, vc bu dks.kksaesalaca/ Kkr djsa tc js[kk,m vkSj nlekarj gSaA vki tkurs gSa fd vkidhvH;kl&iqfLrdk ij cuh lh/h ydhjsa(ruled lines)

ijLij lekarj gksrh gSaA blfy,] bu ydhjksaosQ vuqfn'kiVjh vkSj isafly dh lgk;rk lsnks lekarj js[kk, Hkh[khafp,] tSlk fd vkfr 6-19 esa n'kkZ;k x;k gSA

vkfr6.19

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vc laxr dks.kksaosQ fdlh Hkh ;qXe dks ekfi, vkSj muosQ chp esa laca/ Kkr dhft,Avki Kkr dj ldrs gSa fd 1 = 5, 2 = 6, 4 = 8 vkSj 3 = 7gSA bllsvki fuEu vfHkx`ghr dks Lohr dj ldrsgSa%

vfHkx`ghr6.3 :;fn ,d fr;Zd js[kk nks lekarj js[kkvksa dksizfrPNsn djs] rks laxr dks.kksadk izR; sd ;qXe cjkcj gksrk gSA

vfHkxghr 6-3 dkslaxr dks.k vfHkx`ghr Hkh dgk tkrk gSA vkb, vc blvfHkx`ghr osQ foykse (converse)dh ppkZ djsa] tksfuEu gS%

^;fn ,d fr;Zd js[kk nks js[kkvksa dks bl izdkj izfrPNsn djsfd laxr dks.kksa dk ,d

;qXe cjkcj gks] rks nksuksajs[kk,lekarj gksrh gSa*AD;k ;g dFku lR; gS\ bldh tkp fuEu izdkj dh tk ldrh gS % ,d js[kk AD

[khafp, vkSj ml ij nksfcanqB vkSj Cvafdr dht,AB vkSj Cij e'k% ABQ vkSj BCSdh jpuk dhft, tksijLij cjkcj gksa] tSlk fd vkfr6.20 (i)esan'kkZ;k x;kgSA

vkfr6.20

QB vkSj SCdksADosQ nwljh vksj c

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D;k ge ,d fr;Zd js[kk }kjk nkslekarj js[kkvksadksizfrPNsn djus ls cus ,dkarj var% dks.kksa osQ chp dksbZlaca/ Kkr djus osQ fy, laxr dks.k vfHkx`ghr dkiz;ksx dj ldrs gSa\ vkfr 6-21esa] fr;Zd js[kk PSlekarj js[kkvksa AB vkSjCDdks e'k% fcanqvksaQ vkSjRij izfrPNsn djrh gSA

D;k BQR = QRC vkSj AQR = QRDgSa\

vki tkurs gSafd PQA = QRC (1)

(laxr dks.k vfHkx`ghr)D;k PQA = BQRgS\ gk! (D;ksa\) (2)

blfy,(1) vkSj(2)ls]ge fu"d"kZ fudky ldrs gSafd BQR = QRC

blh izdkj] AQR = QRD

mijksDr ifj.kke dks ,d izes;(theorem)osQ :i esafuEu izdkj ls fy[kk tk ldrk gS%

izes;6.2 :;fn ,d fr;Zd js[kk nks lekarj js[kkvksa dksizfrPNsn dj s] rks,dkarj var% dks.kksadk izR; sd ;qXe cjkcj gksrk gSA

vc] laxr dks.k vfHkx`ghr osQ foykse dk iz;ksx djosQ D;k ge ,dkarj var% dks.kksa osQ,d ;qXe osQ cjkcj gksusij nksuksajs[kkvksadks lekarj n'kkZ ldrsgSa\ vkfr 6-22 esa]fr;Zd js[kk PSjs[kkvksa AB vkSj CDdkse'k% fcanqvksaQ vkSjRij bl izdkj izfrPNsndjrh gS fd BQR = QRCgSA

D;k AB || CDgS?

BQR = PQA (D;ksa?) (1)

ijUrq] BQR = QRC (fn;k gS) (2)

vr%] (1) vkSj (2) lsvki fu"d"kZ fudky ldrs gSafd

PQA = QRC

ijUrq ;s laxr dks.k gSaA

vr%]AB || CDgSA (laxr dks.k vfHkx`ghr dk foykse)

vkfr6.21

vkfr6.22

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bl dFku dks ,d izes; osQ :i esafuEu izdkj O;Dr fd;k tk ldrk gS%izes; 6.3 : ;fn ,d fr;Zd js[kk nksjs[kkvksa dksbl izdkj izfrPNsn djs fd ,dkarj var%dks.kksadk ,d ;qXe cjkcj gS] rksnksuk sa js[kk, ijLij lekarj gksrh gSaA

blh izdkj] vki fr;Zd js[kk osQ ,d gh vksj osQ var% dks.kksa lslacaf/r fuEufyf[krnksizes; izkIr dj ldrs gSa%

izes;6.4 :;fn ,d fr;Zd js[kk nks lekarj js[kkvksa dksizfrPNsn djs] rksfr;Zd js[kk osQ,d gh vksj osQ var% dks.kksa dk izR;sd ;qXe laiwjd gksrk gSA

izes;6.5 :;fn ,d fr;Zd js[kk nksjs[kkvksadksbl izdkj izfrPNsn djsfd fr;Zd js[kk

osQ ,d gh vksj osQ var% dks.kksa dk ,d ;qXe laiwjd gS] rks nksuksa js[kk, ijLij lekarj gksrhgSaAvkidks ;kn gksxk fd bu lHkh vfHkx`ghrksa vkSj izes;ksa dh tkp fiNyh d{kkvksaesavki

oqQN f;kdykiksaosQ }kjk dj pqosQ gSaA vki bu f;kdykiksadks ;gknksgjk ldrsgSaA

6.6,d gh js[kk osQ lekarj js[kk,

;fn nks js[kk, ,d gh js[kk osQ lekarj gksa] rksD;k osijLij lekarj gksaxh\ vkb, bldh tkp djsaAvkfr 6-23 dksnsf[k,] ftlesam || lgSvkSj n|| lgSA

vkb, js[kkvksal, m

vkSjn

osQ fy, ,d fr;Zd js[kkt[khapsaA ;g fn;k gSfd m ||lgS vkSj n ||lgSA

vr%] 1 = 2 vkSj 1 = 3gSA(laxr dks.k vfHkxghr)

blfy,] 2 = 3 (D;ksa\)

ijUrq 2 vkSj 3laxr dks.k gSavkSj cjkcj gSaA

vr%] vki dg ldrs gSa fd

m || n (laxr dks.k vfHkx`ghr dk foykse)

bl ifj.kke dks ,d izes; osQ :i esafuEu izdkj O;Dr fd;k tk ldrk gS%izes;6.6 :os js[kk,tks ,d gh js[kk osQ lekarj gksa] ijLij lekarj gksrh gSaA

fVIi.kh:mijksDr xq.k dks nksls vf/d js[kkvksaosQ fy, Hkh ykxwfd;k tk ldrk gSA

vkb, vc lekarj js[kkvksa ls lacaf/r oqQN iz'u gy djsa%

vko`Qfr 6.23

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mnkgj.k4 :vkfr6.24esa,;fnPQ || RS, MXQ = 135 vkSj MYR = 40gS, rk s XMYKkr dhft,A

vkfr6.24 vkfr6.25

gy:;gkgesamls gksdj] js[kk PQosQ lekarj ,d js[kk AB[khapus dh vko';drk gS]tSlk fd vkfr 6-25 esa fn[kk;k x;k gSA vc] AB || PQ vkSjPQ || RSgSA

vr%] AB || RS gSA (D;ksa\)

vc] QXM + XMB = 180

(AB || PQ,fr;Zd js[kk XMosQ ,d gh vksj osQ var% dks.k)

ijUrq] QXM = 135gSA blfy,]135 + XMB = 180

vr%] XMB = 45 (1)

vc] BMY = MYR (AB || RS,,dkarj dks.k)

vr%] BMY = 40 (2)

(1) vkSj (2) dkstksM+us ij] vkidksizkIr gksxk % XMB + BMY = 45 + 40

vFkkZr] XMY = 85

mnkgj.k5 :;fn ,d fr;Zd js[kk nks js[kkvksa dksbl izdkj izfrPNsn djs fd laxr dks.kksaosQ ,d ;qXe osQ lef}Hkktd ijLij lekarj gksa] rks fl dhft, fd nksuksajs[kk, Hkh ijLijlekarj gksrh gSaA

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gy:vkfr 6-26 esa] ,d fr;Zd js[kkADnksjs[kkvksaPQ vkSjRSdks e'k% fcanqvksaBvkSjCij izfrPNsn djrh gSA fdj.kBE, ABQdh lef}Hkktd gSvkSj fdj.kCG , BCSdh lef}Hkktd gS rFkkBE || CGgSA

geasfl djuk gS fdPQ || RSgSA

;g fn;k gS fd fdj.k BE, ABQdh lef}Hkktd gSA

vr%] ABE =1

2 ABQ (1)

blh izdkj fdj.kCG , BCSdh lef}Hkktd gSA

vr%] BCG =1

2 BCS (2)

ijUrq]BE || CGgS vkSjAD,d fr;Zd js[kk gSA

vr%] ABE = BCG

(laxr dks.k vfHkx`ghr) (3)

(3) esa] (1) vkSj (2) dksizfrLFkkfir djus ij] vkidksizkIr gksxk%

1

2

ABQ =1

2

BCS

vFkkZr~] ABQ = BCS

ijUrq] ;s fr;Zd js[kk AD}kjk js[kkvksaPQ vkSj RSosQ lkFk cuk, x, laxr dks.k gSa vkSj;s cjkcj gSaA

vr%] PQ || RS

(laxr dks.k vfHkx`ghr dk foykse)

mnkgj.k6 : vkfr6.27esa, AB || CD vkSjCD || EFgSAlkFk gh]EA ^ABgSA;fn BEF = 55gS] rksx,yvkSjzosQ eku Kkr dhft,Agy: y+ 55 = 180 (CD || EF,fr;Zd

js[kk EDosQ ,d gh vksj osQ var% dks.k)

vkfr6.26

vkfr6.27

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vr%] y= 180 55 = 125

iqu%] x= y (AB || CD,laxr dks.k vfHkx`ghr)

blfy,] x= 125

vc pwafdAB || CD vkSjCD || EFgS,blfy,AB || EFgSA

vr%] EAB + FEA = 180

(fr;Zd js[kk EAosQ ,d gh vksj osQ var% dks.k)

blfy,] 90 +z+ 55 = 180

ftlls] z= 35izkIr gksrk gSA

iz'ukoyh6.2

1. vkfr6.28esa, xvkSjy osQ eku Kkr dhft, vkSjfiQj n'kkZb, fdAB || CDgSA

2. vkfr 6.29esa] ;fnAB || CD, CD || EF vkSjy:z= 3 : 7gS,rksxdk eku Kkr dhft,A

vkfr6.28

vkfr6.29

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126 xf.kr


3. vkfr 6.30esa] ;fnAB || CD, EF ^ CD vkSj GED = 126gS] rks AGE, GEF vkSj FGE

Kkr dhft,A

4. vkfr6.31esa,;fnPQ || ST, PQR = 110 vkSjRST = 130gS] rks QRSKkr dhft,A

[laosQr %fcanq Rlsgksdj STosQ lekarj ,d js[kk[khafp,A]

5. vkfr6.32esa,;fnAB || CD, APQ = 50 vkSj PRD = 127gS] rks xvkSjyKkr dhft,A

6. vkfr6.33esa]PQ vkSjRSnks niZ.k gSatks,d nwljsosQ lekarj j[ks x, gSaA ,d vkiru fdj.k (incident

ray) AB,niZ.kPQlsBijVdjkrh gSvkSj ijkofrZrfdj.k (reflected ray)iFkBCij pydj niZ.k RSlsCij Vdjkrh gS rFkk iqu%CDosQ vuqfn'k ijkofrZrgkstkrh gSAfl dhft, fd AB || CDgSA

6.7f=kHkqt dk dks.k ;ksx xq.k

fiNyh d{kkvksaesavki f;kdykiksa }kjk ;g lh[k pqosQ gSa fd ,d f=kHkqt osQ lHkh dks.kksadk ;ksx 180gksrk gSA ge bl dFku dkslekarj js[kkvksa lslacaf/r vfHkx`ghrksavkSj izes;ksadk iz;ksx djosQ fl dj ldrsgSaA

vkfr6.30

vkfr6.31

vkfr6.32

vkfr6.33

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izes;6.7 :fdlh f=kHkqt osQ dks.k ksa dk ;ksx 180gksrk gSAmiifk:vkb, ns[ksafd gesamijksDr dFku esaD;k fn;k gS] vFkkZr~gekjh ijfdYiuk (hypothesis)D;k gSvkSj gesa D;k fl djuk gSA gesa,df=kHkqt PQRfn;k gSrFkk 1, 2 vkSj 3blf=kHkqt osQ dks.k gSa(nsf[k, vkfr 6-34)A

gesa] 1 + 2 + 3 = 180fl djuk gSAvkb,

HkqtkQRosQ lekarj mlosQ lEeq[k 'kh"kZPlsgksdj,d js[kk XPY[khapsa] tSlk fd vkfr 6-35 esan'kkZ;k x;k gSA blls ge lekarj js[kkvksa ls lacafkrxq.kksadk iz;ksx dj ldrs gSaA

vc]XPY,d js[kk gSA

vr%] 4 + 1 + 5 = 180gSA (1)

ijUrqXPY || QRrFkkPQ vkSjPRfr;Zd js[kk,gSaA

blfy,] 4 = 2 vkSj 5 = 3(,dkarj dks.kksaosQ ;qXe)

4 vkSj 5osQ ;s eku (1)esa] j[kus ijgesa izkIr gksrk gS%

2 + 1 + 3 = 180

vFkkZr~] 1 + 2 + 3 = 180gSA n

;kn dhft, fd vkius fiNyh d{kkvksaesa] ,d f=kHkqt osQ cfg"dks.kksa (exterior angles)osQckjsesa v;;u fd;k Fkk (nsf[k, vkfr 6-36)A Hkqtk QRdks fcanq Srd c

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128 xf.kr


bl ifj.kke dks ,d izes; osQ :i esafuEu izdkj O;Dr fd;k tk ldrk gS%

izes;6.8 :;fn ,d f=kHkqt dh ,d Hkqtk c

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blh izdkj] fdj.kCOdks.kBCDdh lef}Hkktd gSA

vr%] BCO =1

2 BCD

=1

2(180 z) = 90

2

z(2)

D BOCesa] BOC + BCO + CBO = 180gSA (3)(1) vkSj(2)dks(3)esa j[kus ij] vkidks izkIr gksxk %

BOC + 90

2

z+ 90

2

y= 180

blfy,] BOC =2

z+

2

y

;k] BOC =1

2(y+z) (4)

ijUrq] x+y+z= 180 (f=kHkqt dk dks.k ;ksx xq.k)vr%] y+z= 180 x

blls(4)fuEu gkstkrk gS %

BOC =1

2

(180 x)

= 90 2

x

= 90 1

2 BAC

iz'ukoyh6.3

1. vkfr6.39esa]D PQRdh HkqtkvksaQP vkSj RQdkse'k% fcanqvksaS vkSj Trd c

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130 xf.kr


2. vkfr6.40esa] X = 62 vkSjXYZ = 54gSA;fnYO vkSj ZOe'k %D XYZosQ XYZ vkSj XZY osQ lef}Hkktd gSa] rks OZY vkSj YOZKkr dhft,A

3. vkfr6.41esa];fn AB || DE, BAC = 35 vkSj CDE = 53gS] rks DCEKkr dhft,A

4. vkfr6.42esa];fn js[kk,PQ vkSjRSfcanq Tij blizdkj izfrPNsn djrh gSafd PRT = 40, RPT

= 95 vkSjTSQ = 75gS]rks SQTKkr dhft,A

5. vkfr 6.43esa] ;fn PQ ^ PS, PQ || SR, SQR = 28 vkSj QRT = 65gS] rks xvkSj yosQ

eku Kkr dhft,A

vko`Qfr 6.40

vko`Qfr 6.41

vko`Qfr 6.42

vko`Qfr 6.43

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6. vkfr 6-44 esa] D PQRdh Hkqtk QRdks fcanq Srd c

ganit (6) - रेखाएं और कोण

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