Lagnever AI English

Triangle inequality aur angle sum property

Short mein jawab

Teen lambaaiyon se triangle tabhi banta hai jab har ek baaki do ke jod se kam ho (triangle inequality); sabse lambi jaancho. Do angles se triangle tabhi banta hai jab unka jod 180° se kam ho. Har triangle ke angles ka jod 180° hai, aur exterior angle saamne ke do andar ke angles ke jod jitna hota hai.

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Har teen lambaaiyon se triangle nahi banta

3, 4 aur 8 cm, ya 2, 3 aur 6 cm sides waale triangles banaane ki koshish karo. Kitni bhi saavdhaani se kheencho, dono arcs kabhi nahi milte. Har case ko bana kar jaanchna dheema hoga, isliye hum ek niyam dhoondhte hain.

Seedha raasta ya ghoomkar jaane waala raasta?

Samtal zameen par ek tent, ek ped aur ek khamba socho. Tent se ped tak seedha chalna pehle khambe tak aur phir ped tak jaane se chhota hai. Isi tarah har triangle mein do vertices ko jodne waali side teesre vertex se hokar jaane waale raaste se chhoti hoti hai. Maan lo 10 cm, 15 cm aur 30 cm kisi triangle ki sides hain. Har side ko baaki do se milaao: 10 < 15 + 30 aur 15 < 10 + 30, par 30 to 10 + 15 = 25 se zyaada hai, yaani seedha raasta ghoomkar jaane waale raaste se lamba ho jaata, jo bekaar baat hai. Aisa triangle nahi hota.

Tent, ped aur khamba: seedha raasta khambe se hokar jaane waale raaste se chhota hai
Seedha raasta hamesha chhota

Triangle inequality

Teen lambaaiyan triangle inequality tab poori karti hain jab har ek baaki do ke jod se kam ho. Do chhoti lambaaiyan ye jaanch hamesha paas karti hain, kyunki har ek akele sabse lambi se hi chhoti hai, isliye sirf sabse lambi ko jaanchna padta hai. 4, 5 aur 8 ke liye 8 < 4 + 5 = 9 jaanchna kaafi hai.

Par kya niyam poora hone par triangle hamesha banta hai? Sabse lambi side ko base AB socho, aur A aur B se do chhoti lambaaiyon ke radius waale circles. Radii ka jod theek AB ho to circles bas chhoote hain; kam ho to kabhi nahi milte; aur zyaada ho, jaisa triangle inequality kehti hai, to ve do points par ek doosre ko kaatte hain, aur koi bhi point teesra vertex deta hai. Isliye triangle theek tabhi banta hai jab triangle inequality poori ho. Do sides tay hon to teesri side unke fark aur jod ke beech kuch bhi ho sakti hai, dono ko chhodkar: 3 aur 7 ke liye 4 aur 10 ke beech kuch bhi.

Sabse lambi side ke siron se bane do circles chhoote hain, nahi milte ya kaatte hain, is par ki radii ka jod AB ke baraabar, kam ya zyaada hai
Kya dono circles milte hain?

Do angles se triangle kab banta hai

Do angles aur unke beech ki side di ho, to side kheencho aur uske siron par dono angles banaao. Agar dono angles 90° ya usse zyaada hon, to arms saaf taur par kabhi nahi milti. A par 40° ho, to ∠B badhne ke saath B se nikli arm aur aur daayein jhukti hai; 140° par wo pehli arm ke parallel ho jaati hai, kyunki transversal ki ek hi taraf ke interior angles ka jod 180° hota hai. Isliye do angles se triangle tabhi banta hai jab unka jod 180° se kam ho, aur side ki lambaai se koi fark nahi padta.

Base AB aur A par 40°: 140° par B se nikli arm l ke parallel hai aur use kabhi nahi milti
Arms kab kabhi nahi milti?

Angle sum property

60° aur 70° angles waale triangles alag alag bases par banaao, teesra angle har baar wahi 50° aata hai. Kyun, ye dekhne ke liye ∠B = 50° aur ∠C = 70° waala ∆ABC lo, aur A se BC ke parallel line XY kheencho. Alternate angles se ∠XAB = 50° aur ∠YAC = 70° milta hai. XY par A ke angles milkar ek seedha angle banaate hain, isliye ∠BAC = 180° − 120° = 60°. Yahi daleel har triangle par chalti hai: triangle ke teeno angles ka jod 180° hota hai. Upar ke vertex se parallel line ka ye vichaar Euclid ki The Elements mein milta hai, jo lagbhag 300 BCE mein likhi gayi.

Triangle ABC aur A se BC ke parallel line XY: alternate angles dikhaate hain ki teeno angles ka jod 180° hai
Angles ka jod 180° kyun hai

Exterior angles

Side BC ko C se aage point D tak badhaao. Is badhi hui line aur side AC ke beech ka angle ∠ACD ek exterior angle hai. ∠ACB aur ∠ACD milkar ek seedha angle banaate hain, aur ∠A + ∠B + ∠ACB = 180° hai, isliye exterior angle ∠A + ∠B ke baraabar hai, yaani apne saamne ke do andar ke angles ka jod. ∠A = 50° aur ∠B = 60° ho to exterior angle 110° hai.

Triangle ABC aur BC ko D tak badhaaya gaya: 110° ka exterior angle 50° + 60° ke baraabar hai
Exterior angle

Ye kyun zaroori hai

Ye do niyam kuch bhi kheenchne se pehle bata dete hain ki triangle ban sakta hai ya nahi, aur angle sum property poori geometry mein kaam aati hai. Constructions khud triangle kaise banayein mein hain, aur triangles ke prakaar triangles ke types mein. Har sawaal ke jawaab Chapter 7 ke NCERT solutions mein hain; poore Chapter 7 ke notes bhi dekho.

Practice karo

  1. Kaunsa set triangle inequality poori karta hai?

    1. (a) 2, 2, 5
    2. (b) 3, 6, 9
    3. (c) 5, 10, 12
    4. (d) 1, 1, 5
    Jawab dekho

    Jawab: (c) 5, 10, 12

  2. Jis triangle ke do angles 150° aur 15° hain, uska teesra angle kya hai?

    1. (a) 5°
    2. (b) 15°
    3. (c) 25°
    4. (d) 195°
    Jawab dekho

    Jawab: (b) 15°

  3. Triangle ka ek angle 40° ho to kaunsa doosra angle triangle ko namumkin bana deta hai?

    1. (a) 100°
    2. (b) 120°
    3. (c) 139°
    4. (d) 140°
    Jawab dekho

    Jawab: (d) 140°

  4. Do sides 3 cm aur 7 cm hon to teesri side kin values ke beech honi chahiye?

    Jawab dekho

    Jawab: 4 cm aur 10 cm ke beech, dono ko chhodkar: ye 7 − 3 = 4 se lambi aur 7 + 3 = 10 se chhoti honi chahiye.

  5. ∆ABC mein ∠A = 50° aur ∠B = 60° hai. C par exterior angle nikaalo.

    Jawab dekho

    Jawab: ∠ACB = 180° − 110° = 70°, isliye exterior angle 180° − 70° = 110° hai, jo 50° + 60° bhi hai.

Students ke sawal

Triangle inequality kya hai?

Ye niyam hai ki triangle ki har side baaki do sides ke jod se chhoti hoti hai, kyunki do vertices ke beech seedha raasta teesre vertex se hokar jaane waale raaste se chhota hai. Jo teen lambaaiyan ise poora karti hain unse hamesha triangle banta hai, aur jo ise todti hain unse kabhi nahi.

Teen lambaaiyon se triangle banega ya nahi, ye jaldi kaise pata karein?

Unhe kram mein rakho aur sabse lambi ko baaki do ke jod se milaao. Sabse lambi chhoti ho to triangle banta hai: 4, 5, 8 chalta hai kyunki 8 < 9. Baraabar ya badi ho to triangle nahi banta: 3, 6, 9 fail hai kyunki 9 = 3 + 6, aur arcs bas chhoo kar reh jaate.

Triangle ki angle sum property kya hai?

Kisi bhi triangle ke teeno angles ka jod 180° hota hai. Ek vertex se saamne ki side ke parallel line kheencho: wo jo alternate angles banaati hai ve triangle ke baaki do angles ke baraabar hain, aur us vertex ke angle ke saath milkar ek seedha angle, 180°, banaate hain.

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