Class 6 Maths Chapter 2 NCERT Solutions Hinglish mein: Lines and Angles

Ganita Prakash (Reprint 2026-27) ke Chapter 2, Lines and Angles, ke har “Figure it Out” sawaal ke jawab, in-text sawaalon, protractor activities aur paheli ke saath. Page numbers book se milte hain, aur jawab chapter ke saath chhape NCERT ke apne solutions se milaaye gaye hain. Jahan figure naapna ho, wahan ek-do degree ka fark aam hai. Short notes: Chapter 2 notes.

Figure it Out: points, segments, lines aur rays (pages 15–17)

1. Rihan aur Sheetal. Rihan ke ek point se anginat lines ban sakti hain. Sheetal ke do points se theek ek line banti hai.

2. Fig. 2.4. Line segments hain LM, MP, PQ aur QR. Points L aur R theek ek segment par hain; M, P aur Q do-do segments par.

3. Fig. 2.5. Rays hain TA, TN, TB aur NB (TN aur TB ek hi raaste par ek disha mein hain). TA, TN aur TB ka starting point T hai, par NB ka nahi, wo N se shuru hoti hai.

4. Rough figures:

• a. Do lines banao jo point O par kaatein, ek P se aur ek Q se guzre.
• b. P aur Q se guzarti line banao, aur X se shuru hokar Y se guzarti ray jo line ko point M par kaate.
• c. Points E aur F se guzarti line l banao, aur usse door point D lagao.
• d. Segment AB banao aur A aur B ke beech point P lagao.

5. Fig. 2.6.

• a. Paanch points: D, E, O, B aur C.
• b. Ek line: DB (ise DE, DO, EO, EB ya OB bhi keh sakte ho).
• c. Chaar rays: OC, OB, OE aur OD.
• d. Paanch line segments: DE, DO, DB, EO aur EB (OB aur OC bhi ho sakte hain).

6. B se guzarti ray OA. a. Haan: ye O se shuru hoti hai aur B isi disha mein is par hai, isliye OA aur OB ek hi ray hain. b. Nahi: OA, O se shuru hoti hai, par AO, A se shuru hokar O ki taraf jaayegi, jo alag ray hai.

Figure it Out: angles (pages 19–21)

1. Chitron mein angles. Haan. Cycle mein ek angle ∠BDC hai, vertex D aur arms DB aur DC. Seedhi (ladder), pul aur lakdi ke crate mein bhi jahan do dande ya takhte milte hain wahan angles hain.

2. Arms ST aur SR waala angle. Ek hi point S se rays ST aur SR banao. Angle ∠TSR (ya ∠RST) hai, vertex S.

3. ∠APB ko ∠P kyun nahi keh sakte. P se teen rays shuru hoti hain, isliye teen angles ka vertex P hai: ∠APB, ∠BPC aur ∠APC. ‘∠P’ se pata nahi chalega kaunsa.

4. Nishaan lage angles hain ∠RTQ aur ∠RTP.

5. Teen points A, B, C, ek line par nahi. 3 lines milti hain: AB, BC aur CA. 3 angles ke naam likh sakte ho: ∠ABC, ∠BCA aur ∠CAB.

6. Chaar points, koi teen ek line par nahi. 6 lines milti hain: AB, BC, CD, DA, AC aur BD. 12 angles ke naam likh sakte ho, har point par teen: ∠BAC, ∠CAD, ∠BAD, ∠ABC, ∠ABD, ∠CBD, ∠BCA, ∠ACD, ∠BCD, ∠ADB, ∠ADC aur ∠BDC.

Angles ki tulna (pages 21–23)

Munh kholte jaanwar. Har munh ek angle hai jiska vertex munh ka kona aur arms jabde hain. Chitra se chhote se bade ka ek kram: machhli, magarmachh, saanp, aur chooza, jiski chonch sabse zyaada khuli hai.

Kya do angles ki tulna hamesha aasaan hai? Nahi. 89° aur 91° jaise angles lagbhag ek jaise dikhte hain aur sirf superimposition ya naap se tulna ho sakti hai.

Superimposition aur kahan use karte hain? Line segments, squares, golon aur doosre aakaaron ki tulna mein, jaise ek ruler ya sikka doosre par rakhna.

Figure it Out (page 23):

1. Moda hua kaagaz. Crease EF kinaaron ke saath chaar angles banaati hai: ∠AEF, ∠BEF, ∠DFE aur ∠CFE. Tirchhe mod mein ∠AEF aur ∠CFE, ∠BEF aur ∠DFE se bade hain. Alag tarah modne par alag angles bante hain; kinaare ke parallel seedha mod chaar right angles deta hai.

2. a. ∠AOB bada hai: ∠XOY iske andar hai, kyunki ∠AOB = ∠AOX + ∠XOY + ∠YOB. b. ∠AOB bada hai, kyunki OX iske andar hai. c. Koi nahi: B aur C ek hi ray par hain, isliye ∠XOB = ∠XOC.

3. ∠XOY ya ∠AOB? Yahan ek angle doosre ke andar nahi hai, isliye sirf dekhkar nahi keh sakte: superimpose karo ya naapo. Book ki figure par naapne se ∠XOY lagbhag 105° aur ∠AOB lagbhag 55° aata hai, isliye ∠XOY bada hai.

Do saaras (cranes). Gole ka centre vertex par aur OA ko OX par rakhne se arm OY, ∠AOB ke andar aati hai. Isliye pehla saaras bada angle banaata hai.

Khaas angles (pages 27–29)

Kya OC straight angle ko do barabar angles mein baant sakti hai? Haan. Aise modo ki OB theek OA par aa jaaye; crease OC do barabar angles deti hai jo ek-doosre par bilkul fit hote hain. Har ek right angle hai.

Right angle poore chakkar ka kitna hissa hai? Poore chakkar ka chauthaai (aadhe chakkar ka aadha).

Figure it Out (pages 29–31):

1. Classroom mein right angles. Har rectangle khidki ke kono par 4 right angles hain, aur agar wo panes mein bati ho to zyaada. Darwaaze, blackboard, desk aur kitaab ke cover ke kono par bhi right angles hain.

2. Grid par straight angle. A ko us dot se jodo jo line BA ko A ke aage badhaane par aaye. Pehle grid mein A ki row mein A ke baayein 3 dots mein se koi bhi; doosre grid mein usi tirchhi line par upar ke 2 dots mein se koi bhi.

3. Grid par right angle. Pehle grid mein A ke column mein upar ke 2 ya neeche ke 3 dots mein se koi bhi (5 tareeke). Doosre grid mein A se guzarti us tirchhi line ke 4 dots mein se koi bhi jo AB ko right angle par kaatti hai.

4. Do perpendicular crease. a. 4 right angles milte hain. Dono crease par modo to chaaron angles ek-doosre par bilkul fit hote hain, isliye barabar hain, aur chaar barabar angles poora chakkar banaate hain: har ek chauthaai chakkar, yaani right angle. b. Jaise: koi bhi tirchha mod banaakar crease karo. Mode-mode hi dobaara aise modo ki pehli crease ka ek hissa theek doosre hisse par aa jaaye, aur crease karo. Kholne par doosri crease pehli ke perpendicular hogi.

Angles ko classify karna (pages 31–32)

1. Pichhli figures ke angles. Jaise: Case 1 mein Vidya ki kitaab ka cover acute angle banaata hai aur Case 3 mein, seedha lete hue, straight angle; khidki ke kone right angles hain; Fig. 2.5 ka ∠ATB acute hai; aur numbers waale protractor ki figure mein ∠POS (125°) obtuse hai.

2. Right angle se chhote kuch angles (acute) aur right aur straight angle ke beech ke kuch angles (obtuse) alag-alag dishaon mein banao; kisi bhi taraf mudein, unka type wahi rehta hai.

3. ‘Acute’ aur ‘obtuse’ kyun? Acute ka matlab nukeela aur obtuse ka matlab bhotha. Acute angle ka khulaav patla hota hai, nukeele sire jaisa; obtuse angle right angle se zyaada khulta hai aur bhotha dikhta hai.

4. Triangles mein acute angles. Figures mein 3, 12 aur 21 acute angles hain. Agli figure, sabse chhote triangle ko phir baantkar, mein 30 honge. Har baar 9 badhte hain, kyunki har naya bantwaara 3 naye points jodta hai jinme har ek par 3 acute angles hain.

Angle naapna (pages 32–37)

Barabar hisson mein bata gola (page 34):

Hisse	1	2	3	4	5	6	8	9	10	12
Har angle	360°	180°	120°	90°	72°	60°	45°	40°	36°	30°

Figure it Out, bina numbers ka protractor (page 35): a. ∠KAL = 30°; haan, madhyam aur lambe nishaanon se 5-5 aur 10-10 karke gin sakte ho. b. ∠WAL = 50°. c. ∠TAK = 120°.

Numbers ke do set kyun? Taaki angle kisi bhi taraf ke 0 se padh sako: ek scale daayein 0 se shuru hota hai, doosra baayein 0 se.

Numbers waale protractor ki figure ke angles (page 36):

Angle	Maap	Angle	Maap
∠POQ	35°	∠QOU	145°
∠POR	95°	∠ROS	30°
∠POS	125°	∠ROT	65°
∠POT	160°	∠ROU	85°
∠QOR	60°	∠SOT	35°
∠QOS	90°	∠SOU	55°
∠QOT	125°	∠TOU	20°

∠TOQ wahi angle hai jo ∠QOT, 125°. Jin angles ki ek arm OP par hai unke liye andar ka scale, aur jinki arm OU par hai unke liye bahar ka scale use karo. ∠TOS = 55° − 20° = 35°: haan, ghataana kaam karta hai. Agli figure mein OB, 0 par hai aur OA, 80 se guzarti hai, isliye ∠AOB = 80°.

Apna protractor banao aur Think! (pages 37–40)

Aadha chakkar = 360° ka ½ = 180°. Chauthaai chakkar = 360° ka ¼ = 180° ka ½ = 90°. Chakkar ka aathvaan hissa = 360° ka ⅛ = 180° ka ¼ = 90° ka ½ = 45°, jisse 45° aur 135° par crease aati hai. Ek aur baar aadha modne par 22.5° milta hai.

Think! ∠AOB = ∠BOC = … = ∠HOI = 22.5°, kyunki 180° ke straight angle ko 8 barabar hisson mein moda gaya hai: 180° ÷ 8 = 22.5°.

Figure it Out: naapna (pages 40–43)

1. Teeno angles lagbhag 47°, 23° (∠GHK, jo ∠IHJ hi hai) aur 108° hain.

2. Jawab tumhaari classroom pe nirbhar hain: kitaab ya darwaaze ke kone 90° hain; khula darwaaza ya khidki lagbhag 180° tak koi bhi angle bana sakte hain.

3. ∠IHJ = 42° aur ∠IHJ = 116°. Kaagaz ka protractor inhe theek nahi naap sakta, kyunki uski crease sirf har 22.5° par hain.

4. Nishaan laga reflex angle. Arms ke doosri taraf ka angle naapo, lagbhag 100°, aur ghatao: 360° − 100° = 260°.

5. a. 80° b. 120° c. 60° d. 130° e. 130° f. 60°.

6. ∠BXE = 115°, ∠CXE = 85°, ∠AXB = 65° aur ∠BXC = 30°.

7. ∠PQR = 45°, ∠PQS = 100° aur ∠PQT = 150°.

8. Kaagaz ka craft. Mode hue square ki crease 45°, 90° aur 135° ke angles par milti hain, kyunki square ko diagonal par modne se uske right angle kone aadhe ho jaate hain.

9. Triangle ke angles. Book ki figures par naapne se: (a) lagbhag 45°, 65° aur 70°; (b) lagbhag 54°, 63° aur 63°; (c) lagbhag 31°, 52° aur 97°. Har set ka jod 180° aata hai. Anumaan: kisi bhi triangle ke teeno angles ka jod 180° hota hai.

Mind the Mistake, Mend the Mistake (page 44)

Angle	Kya galat hua	Sahi maap (lagbhag)
∠U = 35°	Koi arm 0 waali line par nahi: ek arm base ke neeche hai. Protractor ghumao taaki ek arm 0 se guzre.	60°
∠V = 80°	Neeche ki arm 10 se guzarti hai, 0 se nahi. Ghatao: 80° − 10°	70°
∠W = 70°	Protractor ka centre arm ke sire par hai, us vertex par nahi jahan arms milti hain.	73°
∠X = 150°	Vertex protractor ke centre par nahi, aur galat scale padha gaya.	22°
∠Y = 120°	Koi arm 0 waali base line par nahi. Protractor ghumao taaki ek arm 0 se guzre.	145°
∠Z = 85°	Ek arm bahar ke scale ke 0 par hai, isliye bahar ka scale padhna chahiye.	105°

Angles kahan hain? (pages 45–46)

1. Ghadi. a. 12 number poore chakkar ke 360° ko 12 barabar hisson mein baantte hain, isliye paas-paas ke do numbers ke beech 360° ÷ 12 = 30° hai. b. 2 baje 60°; 4 baje 120°; 6 baje 180°. c. 3 baje aur 9 baje suiyaan right angle, 90°, banaati hain (9 baje doosri taraf se naapo to 270°).

2. Darwaaza. Haan. Vertex us line par hai jahan darwaaza kabze se frame se juda hai; arms band darwaaza (deewar ke saath) aur khula darwaaza hain.

3. Jhoola. Angle aaraam se seedhe latakte jhoole ki zanjeeron aur us sthiti ki zanjeeron ke beech hai jahan se Vidya jhoolna shuru karti hai. Vertex wo point hai jahan se zanjeerein frame se latakti hain.

4. Khilaune ki dhalaanein. Haan. Har dhalaan ke liye ek arm dhalaan ka kinaara hai, jo dikhta hai, aur doosri arm judne ki jagah par khadi side (ya ek kaalpanik aadi line) hai, jo nahi dikhti. Aadi line se angle jitna bada, dhalaan utni tez aur gendein utni tezi se ludhakti hain.

5. Ghooma hua keeda. Haan. Vertex aadi line par wo point hai jiske chaaron or keeda ghoomta hai; ek arm aadi line ke saath aur doosri ghoomne ke baad keede ke shareer ke saath. Inke beech ka angle ghumaav ki maatra hai.

Angle banaana (pages 46–50)

Figure it Out (page 49), 1. Fig. 2.23. Angles mein ∠CAP, ∠ACD, ∠APL, ∠DLP, ∠RPL, ∠SLP, ∠PRS, ∠LSR, ∠BRS aur ∠CLP hain, aur aur bhi, jaise ∠BAC aur ∠PLS. Har ek ka andaaza lagao, phir naapkar table mein milaao.

2. 110°, 40°, 75°, 112° aur 134° banaana. Ek ray banao, protractor ka centre uske starting point par aur ray ko 0 par rakho, usi scale par 0 se zaroori number tak gino, nishaan lagao aur use vertex se jodo.

3. ∠HIJ ki nakal. Pehle naapo: ye lagbhag 116° hai. Phir ek ray banao, protractor ka centre uske starting point par aur ray ko 0 par rakho, usi scale par 116 par nishaan lagao aur vertex se jodo.

Angles ke types aur maap (pages 50–54)

Figure it Out (pages 51–52), 1. Grids. Pehle grid mein AB daayein taraf hai: A ko A ke column ke daayein ke kisi dot se (A ki row par nahi) jodne se acute angle; A ke column ke baayein ke dot se (A ki row par nahi) jodne se obtuse angle banta hai. Doosre grid mein A se guzarti us tirchhi line se tulna karo jo AB ke right angle par hai: B ki taraf ke dots acute aur doosri taraf ke obtuse angles dete hain. Reflex angle ke liye A ko kisi bhi dot se jodo aur dono arms ke bahar ki taraf ka bada angle nishaan lagao.

2. a. ∠PTR = 30° (acute) b. ∠PTQ = 60° (acute) c. ∠PTW = 102° (obtuse) d. ∠WTP = 258° (reflex).

Let’s Explore (page 53). ∠REB straight angle hai, isliye ∠BET = 180° − 80° = 100°. ∠SER right angle hai, isliye ∠SET = 90° − 80° = 10°.

Figure it Out (pages 53–54):

1. 140°, 82°, 70° aur 35° seedhe protractor se banao. 195°, ek reflex angle, ke liye 360° − 195° = 165° banao aur bahar ki taraf ka bada angle nishaan lagao.

2. Andaaza aur naap. Book ki figures par naapne se: a. lagbhag 45° (acute) b. lagbhag 165° (obtuse) c. lagbhag 120° (obtuse) d. lagbhag 34° (acute) e. lagbhag 99° (obtuse) f. lagbhag 348° (reflex, poora ghoomkar nishaan laga).

3. Teen acute, ek right aur do obtuse angles. Jaise, point O se chaar rays OA, OB, OC aur OD aise banao ki ∠AOB = 30°, ∠BOC = 60° aur ∠COD = 40°. Tab ∠AOB, ∠BOC aur ∠COD acute hain, ∠AOC = 90° right hai, aur ∠BOD = 100° aur ∠AOD = 130° obtuse hain.

4 aur 5. Letters M aur Y.

6. Ashoka Chakra. Paas-paas ke aaron ke beech 360° ÷ 24 = 15°. Do aaron ke beech sabse bada acute angle 75° (5 gap) hai, kyunki 6 gap theek 90° banaate hain.

7. Paheli. Angle ka chaar guna 90° se kam hona chahiye, isliye angle 22.5° se kam hai; paanch guna 90° se zyaada hona chahiye, isliye angle 18° se zyaada hai. Poore degrees mein angle 19°, 20°, 21° ya 22° ho sakta hai.

Revision ke liye padho: point, line segment aur ray mein fark, angle kya hai, angles ke types aur protractor se angle kaise naapein.