Class 7 Maths Part 2 Chapter 1: Geometric Twins (Hinglish Notes)

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Geometric Twins congruence ke baare mein hai: do figures kab ek doosre ki bilkul copy hoti hain, aur copy banane ya check karne ke liye kaunse naap kaafi hain? Chapter ek signboard ke symbol se shuru hota hai, phir triangles pe aata hai, naap ke paanch aise sets dhoondhta hai jo triangle ko poori tarah fix karte hain, aur do aise sets dikhaata hai jo nahi karte. Aakhir mein congruence se isosceles aur equilateral triangles ke angles nikaale jaate hain.

Congruent figures

Ek signboard ka symbol, jo do seedhe arms AB aur BC se bana hai, doosre board pe banana hai. Chhoti figure ko trace kar sakte hain, par bade board ke liye naap lena aasaan hai. Sirf do arm lengths kaafi nahi hain. AB = 4 cm aur BC = 8 cm ke saath arms alag-alag angles pe khul sakte hain, aur har angle se alag shape banta hai. Agar ∠ABC bhi fix kar do, jaise 80°, to sirf ek shape bachta hai.

Same shape aur same size wali figures congruent hoti hain. Ek ki tracing doosri pe ekdum fit hoti hai, aur fit karne ke liye tracing ko ghuma ya palat sakte ho. Circle copy karne ke liye sirf uska radius chahiye; rectangle ke liye uski length aur breadth. Isliye barabar radius wale do circles congruent hain, aur same length aur same breadth wale do rectangles bhi.

Congruent triangles: SSS condition

Meera aur Rabia ko school ke ek bade triangular frame ki cardboard copy banani hai, jo trace karne ke liye bahut bada hai. Wo uski sides naapti hain: 40 cm, 60 cm aur 80 cm. Meera kehti hai ki angles ki zaroorat nahi. Kaagaz pe 4 cm, 6 cm aur 8 cm se check karne ke liye Rabia AB = 6 cm kheenchti hai, aur A ke chaaron taraf 4 cm aur B ke chaaron taraf 8 cm radius ke circles banaati hai. Circles do points, E aur F, pe katte hain, to do triangles bante hain, ΔABE aur ΔABF. Ye AB ke aar-paar ek doosre ki mirror image hain, isliye congruent hain, aur Meera sahi thi.

Ye SSS (Side Side Side) condition hai: agar do triangles ki teeno side lengths same hon, to wo congruent hain. Steps aur picture ke liye padho congruent triangles kya hain.

Congruence kaise likhein

Agar ΔABC, ΔXYZ pe aise fit ho ki A, X pe, B, Y pe aur C, Z pe aaye, to ye corresponding vertices hain. Tab AB aur XY, BC aur YZ, aur CA aur ZX corresponding sides hain, aur ∠A = ∠X, ∠B = ∠Y aur ∠C = ∠Z. Hum likhte hain ΔABC ≅ ΔXYZ, aur letters ka kram batata hai ki kaunsa vertex kisse match karta hai. ΔACB ≅ ΔXZY bhi yahi baat kehta hai, par ΔACB ≅ ΔXYZ galat hai, kyunki ye C ko Y se jodta hai.

Jaise, diagonal BD rectangle ABCD ko ΔABD aur ΔCDB mein kaatta hai. AB = CD aur AD = CB, kyunki ye rectangle ki aamne-saamne ki sides hain, aur BD dono mein common hai, isliye SSS se triangles congruent hain. AB ko CD pe aana hai, isliye A ka match C, B ka D aur D ka B: ΔABD ≅ ΔCDB.

Congruent triangles ki paanch conditions

• SSS: teeno sides barabar.
• SAS: do sides aur included angle, yaani unke beech ka angle. Agar AB = XY = 6 cm, AC = XZ = 5 cm aur ∠A = ∠X = 30°, to jo bhi ye triangle banaaye, use wahi triangle milega, isliye ΔABC ≅ ΔXYZ.
• ASA: do angles aur included side. BC = YZ = 5 cm, ∠B = ∠Y = 50° aur ∠C = ∠Z = 30° triangle ko fix kar dete hain.
• AAS: do angles aur ek side jo unke beech nahi hai. Triangle ke angles milkar 180° hote hain, isliye teesre angles bhi barabar ho jaate hain, aur ASA lag jaata hai. Agar ∠A = ∠X = 35°, ∠C = ∠Z = 75° aur BC = YZ = 4 cm, to ∠B = ∠Y = 180° − 110° = 70°.
• RHS: do right-angled triangles mein hypotenuse (right angle ke saamne ki side) aur ek aur side. BC = YZ = 4 cm, ∠B = ∠Y = 90° aur AC = XZ = 5 cm se construction mein base ke upar ek triangle aur neeche uski mirror image banti hai, aur dono congruent hain.

Jo naap kaafi nahi hain

Teen angles kaafi nahi. 30°, 70° aur 80° angles wale triangles ka shape same hota hai par size kuch bhi ho sakta hai, isliye aise do triangles ka congruent hona zaroori nahi.

Do sides aur non-included angle (SSA) kaafi nahi. PQ = 6 cm, ∠P = 30° aur QR = 4 cm lo. PQ aur P se 30° pe line l kheencho. Q se 4 cm radius ka arc l ko do points, R aur S, pe kaatta hai, isliye ΔPQR aur ΔPQS dono naap pe fit hote hain par saaf alag hain. RHS, SSA ka ek special case hai jo kaam karta hai, kyunki wahan angle right angle hai.

Congruence ka istemaal

Congruence se bina naape saabit kar sakte hain ki lambaaiyan ya angles barabar hain. Agar O, AD aur BC dono ka midpoint hai, to AO = OD aur BO = OC, aur ∠AOB = ∠DOC kyunki ye vertically opposite angles hain. SAS se ΔAOB ≅ ΔDOC, jahan A ka match D, O ka O aur B ka C hai. Isliye AB aur DC corresponding sides hain, aur AB = DC.

Isosceles aur equilateral triangles

ΔABC mein AB = AC aur ∠A = 80° lo. A se BC tak altitude AD kheencho. Triangles ADB aur ADC mein D pe right angle hai, hypotenuse AB aur AC barabar hain, aur AD common side hai, isliye RHS se ΔADB ≅ ΔADC. Isliye ∠B = ∠C, aur har ek (180° − 80°) ÷ 2 = 50° hai. Har triangle mein barabar sides ke saamne ke angles barabar hote hain.

Equilateral triangle mein teeno sides barabar hain, isliye teeno angles barabar hain. Wo milkar 180° hote hain, isliye har angle 60° hai. Poori reasoning aur aur examples isosceles aur equilateral triangle ke angles mein hain.

Hamare aas-paas congruent triangles

Congruent triangles puraane zamaane se aaj tak imaaraton aur designs mein dikhte hain. Inhe Paris ke Louvre Museum ke glass pyramid, Egypt ke Great Pyramid of Giza, domes ke frames, rangoli patterns aur Kolkata ke Rabindra Setu (Howrah Bridge) ke steel frame mein dhoondho. Chapter ke har sawal ka jawab Part 2 Chapter 1 ke NCERT solutions mein hai.