Linear pair aur vertically opposite angles

Jab lines milti hain

Ek chaukor kaagaz ko alag-alag tarah modo aur pencil aur scale se tahon pe line kheencho. Koi bhi do lines chuno: kya ve kaagaz pe milti hain? Agar nahi, to kya kaagaz se aage badhaane pe milengi? Ye sab lines samtal satah pe hain, yaani table, panne ya blackboard jaisi chapti satah pe. Jab samtal satah pe do lines ek bindu pe milti hain, to kehte hain ve intersect karti hain. Do seedhi lines sirf ek bindu pe kaat sakti hain: kaatne ke baad ve ek doosre se door hoti jaati hain.

Kaatne ki jagah chaar angles

Jahan line l, line m ko kaatti hai, wahan chaar angles bante hain; bindu ke chaaron or ghoomte hue inhe ∠a, ∠b, ∠c aur ∠d kaho. Activity 1 mein kaatti lines ki kai jodiyan kheencho aur protractor se chaaron angles naapo. Ek pattern dikhta hai: aamne-saamne ke angles barabar aate hain, aur padosi angles ka jod 180° aata hai.

Kya bina naape angles bataaye ja sakte hain? Maan lo ∠a 120° hai. ∠a aur ∠b milkar 180° ka seedha angle banaate hain, isliye ∠b = 180° − 120° = 60°. Isi tarah ∠b aur ∠c seedha angle banaate hain, isliye ∠c = 120°, aur ∠c aur ∠d bhi, isliye ∠d = 60°. To ∠a aur ∠c dono 120° hain, aur ∠b aur ∠d dono 60°.

Linear pair

Kaatne ki jagah ∠a aur ∠b jaise padosi angles ko linear pair kehte hain. Inki baahar ki bhujaayein ek seedhi line pe hoti hain, isliye linear pair ka jod hamesha 180° hota hai. Ek kaatne ki jagah chaar linear pair hote hain: ∠a ke saath ∠b, ∠b ke saath ∠c, ∠c ke saath ∠d, aur ∠d ke saath ∠a.

Vertically opposite angles aur ek proof

Kaatne ke bindu ke aar-paar aamne-saamne ke angles, jaise ∠b aur ∠d, vertically opposite angles hain. Kya ye hamesha barabar hote hain, angles kuch bhi hon? Kai maan aazmaana madad karta hai, par har haal nahi dekh sakta. Tark dekh sakta hai. Seedha angle 180° ka hota hai, isliye ∠a + ∠b = 180° aur ∠a + ∠d = 180°. Dono 180° hain, isliye ∠b aur ∠d barabar hone chahiye. Isi tarah ∠b + ∠a = 180° aur ∠b + ∠c = 180°, isliye ∠a = ∠c. To kinhi bhi do kaatti lines ke vertically opposite angles barabar hote hain. Aisa tark jo bina naape har haal mein chale, proof kehlaata hai.

Jab chaaron angles barabar hon

Agar kaatne ki jagah chaaron angles barabar hon, to har ek 360° ka chauthai, yaani 90°, hai. Aisi lines samkon pe milti hain aur perpendicular lines kehlaati hain; kone pe chhota square banaate hain.

Naap aur geometry

Naapne pe linear pair 179° ya 181° aa sakta hai, ya do vertically opposite angles thode alag aa sakte hain. Iske do kaaran hain: protractor use karne ki galti, aur hamaari kheenchi lines ki motaai. Geometry ki aadarsh line ki koi motaai nahi hoti, par aisi line hum kheench nahi sakte. Geometry mein hum aadarsh lines aur aakaaron ke baare mein tark karte hain, isliye jaante hain ki do hisson mein bata seedha angle theek 180° jodta hai. Asli naap in anumaanon ke bahut kareeb aate hain, isiliye geometry physics, kala, engineering aur architecture mein itni zyaada kaam aati hai.

Ye kyun zaroori hai

Linear pair aur vertically opposite angles is chapter ke baaki saare angle niyamon ke auzaar hain. Kabhi na milne waali lines parallel aur perpendicular lines mein samjhaai gayi hain, aur do lines ko kaatne waali line ke angles transversal ke angles mein. Har sawaal ke jawaab Chapter 5 ke NCERT solutions mein hain; poore Chapter 5 ke notes bhi dekho.