Transversal ke angles: corresponding aur alternate

Transversal kya hai?

Kabhi-kabhi aankh se batana mushkil hota hai ki do lines parallel hain ya nahi. Transversal ye tay karta hai. Do lines l aur m ko kaatne waali line t ek transversal hai, aur ye aath angles banaati hai: l pe ∠1 se ∠4, aur m pe ∠5 se ∠8. Kya aathon alag ho sakte hain? Nahi. Har kaatne ki jagah vertically opposite angles barabar hote hain, isliye ∠1 = ∠3, ∠2 = ∠4, ∠5 = ∠7 aur ∠6 = ∠8: chaar barabar jodiyan, aur zyaada se zyaada chaar alag naap.

Corresponding angles

Transversal l ke saath angles ka ek set banaati hai aur m ke saath doosra. Dono sets mein ek hi sthaan ke angles, jaise ∠1 aur ∠5, ∠2 aur ∠6, ∠3 aur ∠7, aur ∠4 aur ∠8, corresponding angles hain.

Activity 3 mein aisi lines aur transversal chahiye jo sirf do alag angles banaayein. l aur transversal t ko X pe milte hue kheencho, aur maano X pe angle 60° hai; uska linear pair saathi 120° hai. t pe bindu Y lo aur Y se aisi line m kheencho jo t ke saath bhi 60° banaaye, angle ko tracing paper ya protractor se copy karke. Lines l aur m parallel aati hain. To jab transversal ke corresponding angles barabar hon, lines parallel hoti hain.

Ulta bhi sach hai. Agar transversal do parallel lines ko kaatta hai, to ek angle tracing paper pe utaaro aur corresponding angle pe khisko: vo bilkul mel khaata hai. Parallel lines ko kaatne waale transversal ke corresponding angles hamesha barabar hote hain. Aur agar do lines parallel nahi hain, jaise Activity 5 mein, to koi bhi transversal unki corresponding jodi barabar nahi bana sakta.

Alternate angles

Ab parallel lines l aur m aur transversal t lo, jinke angles l pe a, b, c, d aur m pe e, f, g, h hain. ∠f aur ∠d alternate angles ki jodi hain, aur ∠e aur ∠c bhi: ye lines ke beech transversal ki ulti taraf hain. Agar ∠f 120° hai, to ∠d kitna? ∠b, ∠f ka corresponding hai, isliye ∠b = 120°, aur ∠d, ∠b ka vertically opposite hai, isliye ∠d = 120°. Yahi kadam ∠f ke kisi bhi naap pe chalte hain, isliye parallel lines ko kaatne waale transversal ke alternate angles hamesha barabar hote hain.

Ek hi taraf ke interior angles

Example 3 mein parallel lines ko transversal kaatta hai aur ∠3 50° hai. Tab ∠2 = 180° − 50° = 130°, ∠3 ke saath linear pair, aur corresponding hone se ∠6 = ∠2 = 130°. ∠3 aur ∠6 jaise angles, jo lines ke beech transversal ki ek hi taraf hain, interior angles hain, aur 50° + 130° = 180°. Ye hamesha hota hai: ∠2 + ∠3 = 180° aur ∠2 = ∠6, isliye ∠3 + ∠6 = 180°.

Angle niyamon ka upyog

Ek angle pata ho to baaki nikal aate hain. Example 1 mein ∠6 135° hai: tab corresponding aur vertically opposite angles se ∠2, ∠4 aur ∠8 bhi 135° hain, aur baaki chaar 180° − 135° = 45° hain. Example 2 mein ∠a 120° aur ∠f 70° hai; tab ∠b = 60°, par ∠b aur ∠f corresponding angles hain aur barabar nahi, isliye lines parallel nahi hain.

Example 4 mein ABCD mein AB, CD ke parallel hai aur AD, BC ke parallel, saath mein ∠DAC = 65° aur ∠ADC = 60°. AB aur CD ko kaatne waale transversal AD se interior angles ∠DAB = 180° − 60° = 120° dete hain, isliye ∠CAB = 120° − 65° = 55°. AD aur BC ko kaatne waale transversal CD se ∠BCD = 180° − 60° = 120°, aur isi tarah ∠ABC = 60°.

Ye kyun zaroori hai

Transversal parallel lines ki pakki jaanch deta hai, aur ek angle se kai angles bata deta hai. Ek kaatne ki jagah ke angles linear pair aur vertically opposite angles mein hain, aur parallel lines banaana parallel aur perpendicular lines mein. Chhahon Figure it Out sawaal Chapter 5 ke NCERT solutions mein hal kiye gaye hain; poore Chapter 5 ke notes bhi dekho.