Congruent triangles kya hain?
Short mein jawab
Do figures congruent hoti hain jab unka shape aur size ek jaisa ho, yaani ek doosri pe fit ho, ghumane ya palatne ke baad bhi. Do triangles SSS, SAS, ASA, AAS ya RHS se match karein to congruent hain. Tab corresponding sides aur angles barabar hain, aur ΔABC ≅ ΔXYZ ka kram batata hai ki kaun kisse match karta hai.
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Congruent ka matlab
Do figures congruent hoti hain jab unka shape bhi same ho aur size bhi same. Test hai unhe superimpose karna: ek ko trace karo ya kaat lo, aur doosri pe rakho. Agar ekdum fit ho jaaye, to figures congruent hain. Rakhne se pehle tracing ko ghuma sakte ho ya page ki tarah palat sakte ho, kyunki dono se na shape badalta hai na size.
Badi figures ko trace karna mushkil hai, isliye hum naap lete hain. Circle apne radius se fix hota hai, aur rectangle apni length aur breadth se. Do arms se bani figure ke liye do lambaaiyan kaafi nahi; arms ke beech ka angle bhi chahiye.
Same teen sides: SSS
Maan lo ek triangle ki sides 4 cm, 6 cm aur 8 cm hain. AB = 6 cm kheencho. Teesra vertex A se 4 cm aur B se 8 cm door hona chahiye, isliye wo A ke chaaron taraf 4 cm radius ke circle pe bhi hai aur B ke chaaron taraf 8 cm radius ke circle pe bhi. Dono circles do points pe katte hain, AB ke upar E aur neeche F.
ΔABE aur ΔABF, AB ke aar-paar mirror images hain, isliye ek ka cut-out palatne ke baad doosre pe fit ho jaata hai. In teen sides wala har triangle usi ek triangle ki copy hai. Yahi SSS condition hai: agar ek triangle ki teen sides doosre ki teen sides ke barabar hon, to triangles congruent hain.
Congruence sahi kram mein likhna
Do congruent triangles ko ek doosre pe rakhne par jo parts aapas mein mil jaate hain, wo corresponding parts hain. Agar A, X pe aaye, B, Y pe aur C, Z pe, to hum likhte hain ΔABC ≅ ΔXYZ. Kram hi sandesh hai:
| Corresponding | Jodiyan |
|---|---|
| vertices | A aur X, B aur Y, C aur Z |
| sides | AB aur XY, BC aur YZ, CA aur ZX |
| angles | ∠A aur ∠X, ∠B aur ∠Y, ∠C aur ∠Z |
Kisi bhi vertex se shuru kar sakte ho, bas dono naam saath-saath badlein: ΔBCA ≅ ΔYZX aur ΔACB ≅ ΔXZY bhi sahi hain, par ΔACB ≅ ΔXYZ nahi. Sahi kram dhoondhne ke liye barabar sides dekho. Diagonal BD wale rectangle ABCD mein AB = CD, AD = CB aur BD common hai. AB ko CD pe aana hai, isliye congruence hai ΔABD ≅ ΔCDB.
Baaki conditions: SAS, ASA, AAS aur RHS
- SAS: do sides aur unke beech ka angle. AB = XY = 6 cm, AC = XZ = 5 cm aur ∠A = ∠X = 30° se sirf ek triangle banta hai, isliye ΔABC ≅ ΔXYZ.
- ASA: do angles aur unke beech ki side, jaise BC = YZ = 5 cm, ∠B = ∠Y = 50° aur ∠C = ∠Z = 30°.
- AAS: do angles aur ek side jo unke beech nahi. Teesre angles barabar ho jaate hain (teeno milkar 180° hote hain), isliye ye ASA ban jaata hai.
- RHS: right-angled triangles ke liye right angle, hypotenuse aur ek aur side, jaise ∠B = ∠Y = 90°, BC = YZ = 4 cm aur AC = XZ = 5 cm.
AAA aur SSA kaafi nahi
30°, 70° aur 80° angles wale triangles kai size mein ban sakte hain, isliye barabar angles se triangles congruent nahi ho jaate. Do sides aur ek aisa angle jo unke beech nahi, wo bhi fail ho sakta hai: PQ = 6 cm, ∠P = 30° aur QR = 4 cm mein Q se 4 cm ka arc 30° wali line ko do points pe kaatta hai, aur do alag triangles bante hain.
Ye kyun zaroori hai
Jab pata ho ki do triangles congruent hain, to har corresponding jodi barabar hoti hai, isliye bina naape saabit kar sakte ho ki do lambaaiyan ya do angles barabar hain. Agla kadam hai isosceles aur equilateral triangle ke angles. Textbook ke sawaalon ke jawab Part 2 Chapter 1 ke NCERT solutions mein hain; poore chapter ke liye Part 2 Chapter 1 ke notes.
Practice karo
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Do triangles ki sides 5 cm, 6 cm aur 7 cm hain. Wo:
Jawab dekho
Jawab: (a) SSS se congruent hain
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ΔDOG ≅ ΔCAT. ∠G kis angle ke barabar hai?
Jawab dekho
Jawab: (c) ∠T G, DOG ka teesra letter hai aur T, CAT ka teesra letter.
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AB = PQ, ∠A = ∠P aur AC = PR. Kaunsi condition dikhati hai ki ΔABC ≅ ΔPQR?
Jawab dekho
Jawab: (b) SAS ∠A, AB aur AC ke beech hai, aur ∠P, PQ aur PR ke beech, isliye barabar angle included angle hai.
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Teen barabar angles se do triangles ka congruent hona kyun saabit nahi hota?
Jawab dekho
Jawab: Bahut se triangles ke teen angles same hote hain, jaise 30°, 70° aur 80°, par wo chhote ya bade ho sakte hain. Unka shape same hai par size hamesha same nahi.
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Do triangles mein ∠A = ∠X = 40°, ∠B = ∠Y = 60° aur BC = YZ hai. Kya wo congruent hain? Condition batao.
Jawab dekho
Jawab: Haan. Teesre angles bhi barabar hain: ∠C = ∠Z = 180° − 100° = 80°. BC, ∠B aur ∠C ke beech hai, aur YZ, ∠Y aur ∠Z ke beech, isliye ASA lagta hai (diye gaye facts AAS hain).
Students ke sawal
Congruent figures kya hoti hain?
Wo figures jinka shape aur size bilkul ek jaisa ho. Ek ki tracing ya cut-out doosri pe ekdum fit hota hai, aur fit karne ke liye use ghuma ya palat sakte hain.
Class 7 mein triangles ki congruence ki conditions kya hain?
SSS (teen sides), SAS (do sides aur unke beech ka angle), ASA (do angles aur unke beech ki side), AAS (do angles aur ek side jo unke beech nahi) aur RHS (right-angled triangles mein hypotenuse aur ek side).
SSA congruence ki condition kyun nahi hai?
Do sides aur ek aise angle se jo unke beech nahi hai, teesre vertex ka arc line ko do points pe kaat sakta hai. Isse same naap wale do alag triangles bante hain, isliye unka congruent hona zaroori nahi.
ΔABC ≅ ΔXYZ mein letters ke kram ka kya matlab hai?
Pehla, doosra aur teesra letter aapas mein match karte hain: A ka X, B ka Y aur C ka Z. Isliye AB = XY, BC = YZ, CA = ZX, aur matching angles barabar hain.
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