NCERT Exemplar ke “Understanding Quadrilaterals” aur “Practical Geometry” units ke topics pe poori practice, ek hi post mein: polygon ke prakar, interior aur exterior angles ka jod, regular polygon, parallelogram, rectangle, rhombus, square, kite aur trapezium ki properties, aur ruler-compass se quadrilateral banane (construction) ke niyam. Pehle chhota recap (saare naye shabd wahin samjhaaye hain), phir 10 MCQs, fill in the blanks, true/false, short aur long answer, HOTS, aur aakhir mein answer key. Har sawaal ka hal Hinglish mein step by step hai. Yeh post apne current chapter Book 1 Ch 4: Quadrilaterals ke saath padhiye: chapter se samjho, yahan se practice karo. 🔷
🧠 Concept Recap: Pehle Yeh Yaad Kar Lo
📖 Pehle Book 1 Ch 4 padho: Quadrilaterals (hub)
📏 Pehle yeh shabd samjho (har shabd yahin samjhaaya hai, aage yahi use honge):
- Polygon (bahubhuj) = sirf seedhi lines (line segments) se bani band shape, jisme lines ek-doosre ko sirf kono pe milti hain. Triangle (3 sides), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8). Gola (circle) polygon nahi, kyunki usme seedhi line nahi.
- Side (bhuja) = polygon ki ek seedhi line. Vertex (sheersh, kona) = jahan do sides milti hain; bahuvachan vertices.
- n = polygon ki sides ki ginti (koi bhi poora number, jaise triangle mein n = 3, pentagon mein n = 5). Formula mein n ki jagah sides ki ginti rakh do.
- ÷ (bhaag) = baraabar hisson mein baantna, jaise 12 ÷ 3 = 4 (12 ko 3 barabar hisson mein baanto, har hissa 4).
- > (bada) = “se bada hai”, jaise 11 > 5 padho “11, 5 se bada hai”.
- Quadrilateral (chaturbhuj) = 4 sides, 4 vertices aur 4 angles wala polygon, jaise copy ka panna, carrom board, patang.
- Angle (kon) aur degree (°) = do lines ke beech ka khulaav. Isse degree mein naapte hain, chinh “°”. Poora chakkar = 360°, seedhi line = 180°, kitaab ke kone jaisa = 90° (right angle, samkon). ∠A padho “angle A” (kona A pe bana angle).
- Interior angle (andar ka kon) = polygon ke andar, ek vertex pe bana angle. Exterior angle (bahar ka kon) = ek side ko aage badhao; badhi hui line aur agli side ke beech ka bahar wala angle. Ek hi vertex pe interior + exterior = 180°, kyunki dono milkar seedhi line banate hain (ise linear pair kehte hain).
- Reflex angle = 180° se bada (par 360° se chhota) angle.
- Convex aur concave = convex polygon mein har interior angle 180° se chhota hota hai (koi kona andar ko dhansa nahi). Concave mein kam se kam ek interior angle reflex (180° se bada) hota hai, jaise teer ke sire (arrowhead) jaisi shape. Is post mein jab tak alag se na kaha jaaye, polygon convex maano.
- Diagonal (vikarn) = do aise vertices ko jodne wali seedhi line jo padosi nahi hain (side nahi hai). Quadrilateral ABCD mein AC aur BD diagonals hain.
- Adjacent (padosi) aur opposite (saamne wale) = ABCD mein AB aur BC padosi sides hain (ek vertex B share karti hain); AB aur CD saamne ki sides hain. Isi tarah ∠A aur ∠B padosi angles, ∠A aur ∠C saamne ke angles.
- Parallel (samaantar), chinh ∥ = do seedhi lines jo kitna bhi badhao, kabhi nahi milti, jaise rail ki do patriyaan. AB ∥ DC padho “AB parallel hai DC ke”.
- Perpendicular, chinh ⊥ = do lines jo 90° pe milti hain, jaise “+” ka nishaan.
- Bisect (aadha-aadha karna) = kisi line ya angle ko do barabar hisson mein baantna. “Diagonals ek-doosre ko bisect karte hain” matlab jahan woh milte hain, woh point dono diagonals ka beech ka point (midpoint) hai.
- Supplementary angles = do angles jinka jod 180° ho, jaise 70° aur 110°.
- Regular polygon = jiski saari sides barabar aur saare angles barabar hon, jaise square ya bilkul barabar hexagon (madhumakkhi ke chhatte ka khaana).
- Perimeter (parimaap) = saari sides ki lambaai ka jod (shape ke kinare-kinare ek chakkar).
- Ratio (anupaat) = 3 : 4 padho “3 is to 4”: matlab cheez ko hisson mein baanta, pehle ko 3 hisse, doosre ko 4 hisse (sab hisse barabar size ke).
- Construction (rachna) = sirf ruler (scale), compass (parkaar) aur protractor (chaandaa, angle naapne wala) se shape ko sahi naap ke saath banana. Arc (chaap) = compass se bana gol line ka tukda. Unique = sirf ek hi tarah ki shape ban sake (koi doosri alag shape nahi).
- Letters jaise x = kisi anjaan number ki jagah rakha “khaali dabba”. 3x matlab 3 guna x (3 × x).
Worked example (pehle ek aasaan sawaal): triangle ke teen angles 50°, 60° aur 70° hain. Jod: 50 + 60 + 70 = 180. Har triangle ke angles ka jod 180° hota hai (ise hum aage use karenge).
1. Quadrilateral ke angles ka jod = 360° (kyun?)
Quadrilateral ABCD mein ek diagonal AC kheecho. Shape do triangles mein bant jaati hai: ABC aur ACD. Har triangle ke angles ka jod 180° hai, aur in do triangles ke saare angles milkar hi quadrilateral ke chaaron angles banate hain. Isliye jod = 180° + 180° = 360°.
2. Kisi bhi polygon ke angles
| Baat | Niyam (shabdon mein) | Example |
|---|---|---|
| Interior angles ka jod | Ek vertex se saare diagonal kheecho: n sides wala polygon (n − 2) triangles mein bantta hai. Jod = (n − 2) × 180° | Pentagon (n = 5): (5 − 2) × 180 = 3 × 180 = 540° |
| Exterior angles ka jod | Convex polygon ke saare exterior angles (har vertex pe ek) ka jod hamesha 360°: kinare-kinare chalo to ek poora chakkar ghoom jaate ho | Triangle ho ya 100 sides wala: 360° |
| Regular polygon ka har exterior angle | 360° ko n barabar hisson mein baanto: 360 ÷ n | Regular hexagon: 360 ÷ 6 = 60° |
| Regular polygon ka har interior angle | 180° − exterior angle (linear pair) | Regular hexagon: 180 − 60 = 120° |
| Sides nikaalna | n = 360 ÷ (ek exterior angle) | Exterior 45° → 360 ÷ 45 = 8 sides (octagon) |
| Diagonals ki ginti | Har vertex se (n − 3) diagonals (khud ko aur do padosiyon ko chhod kar); har diagonal do baar gina gaya, isliye 2 se bhaag: n × (n − 3) ÷ 2 | Hexagon: 6 × 3 ÷ 2 = 9 |
3. Khaas quadrilaterals ki family
| Naam | Pehchaan (definition) | Khaas properties |
|---|---|---|
| Trapezium | Kam se kam ek jodi saamne ki sides parallel (kuch kitaabein “sirf ek jodi” kehti hain) | Parallel sides ke beech wali har tirchhi side ke dono angles ka jod 180° |
| Kite | Padosi sides ki do jodi barabar (jaise AB = AD aur CB = CD) | Diagonals ek-doosre pe 90° (perpendicular); ek diagonal doosre ko aadha karta hai; jin do kono pe alag-alag lambaai ki sides milti hain, woh do angles barabar |
| Parallelogram | Dono jodi saamne ki sides parallel | Saamne ki sides barabar; saamne ke angles barabar; padosi angles ka jod 180°; diagonals ek-doosre ko bisect karte hain |
| Rectangle | Chaaron angles 90° (yeh parallelogram bhi hai) | Parallelogram ki saari properties + diagonals barabar |
| Rhombus | Chaaron sides barabar (yeh parallelogram bhi hai) | Parallelogram ki saari properties + diagonals ek-doosre pe 90° |
| Square | Chaaron sides barabar aur chaaron angles 90° | Rectangle aur rhombus dono ki properties: diagonals barabar, 90° pe, aur ek-doosre ko aadha |
Family ka niyam: square ek khaas rectangle bhi hai aur khaas rhombus bhi. Har rectangle aur har rhombus ek parallelogram hai. Par ulta zaroori nahi: har rectangle square nahi hota.
4. Practical geometry: quadrilateral kab “unique” banta hai?
Triangle ke liye 3 sahi naap kaafi hain, par quadrilateral ke liye 5 naap chahiye. Sirf 4 sides do, to shape “hil” sakti hai (jaise chaar dandiyon ka dhaancha dabaane pe tirchha ho jaata hai), isliye alag-alag shapes ban jaati hain. Yeh 5 naap kuch is tarah ho sakte hain:
- 4 sides + 1 diagonal
- 3 sides + 2 diagonals
- 2 padosi sides + 3 angles
- 3 sides + unke beech ke 2 angles
Khaas shapes ko kam naap chahiye, kyunki unki properties baaki kaam kar deti hain: square ke liye 1 (side), rectangle ke liye 2 (lambaai, chaudaai), rhombus ke liye 2 (side + ek angle, ya dono diagonals), parallelogram ke liye 3 (do padosi sides + beech ka angle).
Triangle wala jaanch-niyam: diagonal shape ko triangles mein baant deta hai. Triangle tabhi banta hai jab kisi bhi do sides ka jod teesri side se bada ho. Agar 3 cm, 4 cm aur 8 cm se triangle banana ho: 3 + 4 = 7, jo 8 se chhota hai, to triangle nahi banega (dono chhoti lines milengi hi nahi).
— 📌 charumam.com
📝 Section A: 10 MCQs
Har sawaal ke 4 options mein se ek sahi chuno. Pehle khud socho, phir “Hal dekho” kholo.
Q1. Ek quadrilateral ke teen angles 75°, 90° aur 105° hain. Chautha angle kitna hai?
Hal (step by step) dekho
Sahi jawab: (c) 90°
Step 1: quadrilateral ke chaaron angles ka jod 360° hota hai.
Step 2: diye gaye teen angles jodo: 75 + 90 + 105 = 270.
Step 3: chautha angle = 360 − 270 = 90°.
Check: 75 + 90 + 105 + 90 = 360 ✅
Q2. Regular hexagon (6 barabar sides aur barabar angles) ka har interior angle kitna hai?
Hal (step by step) dekho
Sahi jawab: (d) 120°
Step 1: regular polygon ka har exterior angle = 360 ÷ (sides). Hexagon mein 6 sides: 360 ÷ 6 = 60°.
Step 2: ek vertex pe interior + exterior = 180° (linear pair). To interior = 180 − 60 = 120°.
Doosra tareeka: kul jod (6 − 2) × 180 = 4 × 180 = 720°, aur 6 barabar angles: 720 ÷ 6 = 120° ✅
Q3. Kis quadrilateral ke diagonals hamesha ek-doosre ko 90° pe kaatte hain, par unka barabar hona zaroori nahi?
Hal (step by step) dekho
Sahi jawab: (b) Rhombus
Rhombus mein chaaron sides barabar hoti hain, aur iske diagonals ek-doosre ko 90° pe aadha-aadha karte hain. Par diagonals ki lambaai alag ho sakti hai (patang jaisi chapti shape).
Rectangle ke diagonals barabar hote hain par 90° pe zaroori nahi. Trapezium aur aam parallelogram mein 90° zaroori nahi.
Q4. Ek regular polygon ka har exterior angle 40° hai. Usme kitni sides hain?
Hal (step by step) dekho
Sahi jawab: (b) 9
Step 1: saare exterior angles ka jod 360° hai, aur regular polygon mein sab barabar hain.
Step 2: sides = 360 ÷ 40 = 9 (ise nonagon kehte hain).
Check: 9 × 40 = 360 ✅
Q5. Parallelogram ABCD mein ∠A = 65° hai. ∠B (∠A ka padosi angle) kitna hai?
Hal (step by step) dekho
Sahi jawab: (b) 115°
Step 1: parallelogram mein padosi angles ka jod 180° hota hai (kyunki AD ∥ BC, aur AB in dono ko kaatne wali line hai; aisi line ke ek hi taraf ke andar wale angles ka jod 180° hota hai).
Step 2: ∠B = 180 − 65 = 115°.
Dhyaan: 65° to ∠C hoga (saamne ka angle), ∠B nahi.
Q6. Octagon (8 sides) mein kitne diagonals hote hain?
Hal (step by step) dekho
Sahi jawab: (b) 20
Step 1: har vertex se diagonal sirf un vertices tak jaata hai jo na woh khud hai, na uske do padosi. To har vertex se 8 − 3 = 5 diagonals.
Step 2: 8 vertices × 5 = 40, par har diagonal do baar gina gaya (AC aur CA ek hi line). To 40 ÷ 2 = 20.
Formula se: 8 × (8 − 3) ÷ 2 = 8 × 5 ÷ 2 = 20 ✅
Q7. Ek aam quadrilateral ko unique (sirf ek tarah ka) banane ke liye kam se kam kitne sahi naap (measurements) chahiye?
Hal (step by step) dekho
Sahi jawab: (c) 5
Quadrilateral ke liye 5 naap chahiye, jaise 4 sides + 1 diagonal. Sirf 4 sides se shape hil kar alag-alag ban jaati hai. Diagonal shape ko do triangles mein baant ke “jakad” deta hai, kyunki 3 sides wala triangle hil nahi sakta.
Q8. Pentagon (5 sides) ke interior angles ka jod kitna hai?
Hal (step by step) dekho
Sahi jawab: (b) 540°
Step 1: ek vertex se diagonal kheecho: pentagon 5 − 2 = 3 triangles mein bantta hai.
Step 2: jod = 3 × 180 = 540°.
Q9. Concave polygon ki pehchaan kya hai?
Hal (step by step) dekho
Sahi jawab: (c) Kam se kam ek interior angle 180° se bada
Concave polygon mein kam se kam ek kona andar ko dhansa hota hai; wahan ka interior angle 180° se bada (reflex) hota hai, jaise teer ke sire (arrowhead) wali shape. Convex mein har interior angle 180° se chhota.
Q10. Rectangle ABCD ke diagonals AC aur BD point O pe milte hain. AC = 12 cm hai. OB kitna hai?
Hal (step by step) dekho
Sahi jawab: (c) 6 cm
Step 1: rectangle ke diagonals barabar hote hain, to BD = AC = 12 cm.
Step 2: diagonals ek-doosre ko aadha (bisect) karte hain, to O, BD ka beech ka point hai: OB = 12 ÷ 2 = 6 cm.
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✍️ Section B: Fill in the Blanks
Khaali jagah (_____) mein sahi shabd ya number bharo.
Q1. Hexagon mein _____ sides hoti hain.
Jawab aur kaaran
Jawab: 6
Hexa = 6. Hexagon ki 6 sides aur 6 vertices hote hain (madhumakkhi ke chhatte ke khaane jaisa).
Q2. Kisi bhi convex polygon ke exterior angles ka jod _____ hota hai.
Jawab aur kaaran
Jawab: 360°
Polygon ke kinare-kinare chalo aur har kone pe mudo: aakhri tak pahunchte-pahunchte aap ek poora chakkar ghoom jaate ho, aur poora chakkar = 360°. Sides kitni bhi hon, jod 360° hi rehta hai.
Q3. Parallelogram ke saamne ke angles _____ hote hain.
Jawab aur kaaran
Jawab: barabar
Parallelogram ABCD mein ∠A = ∠C aur ∠B = ∠D. Jaise ∠A = 70° ho to ∠C bhi 70°.
Q4. Square ke diagonals ek-doosre ko _____° pe kaatte hain.
Jawab aur kaaran
Jawab: 90
Square ek khaas rhombus bhi hai, aur rhombus ke diagonals 90° pe milte hain. To square ke diagonals bhi perpendicular (90°) hote hain, aur barabar bhi.
Q5. Rhombus ki ek side 7 cm hai. Uska perimeter _____ hai.
Jawab aur kaaran
Jawab: 28 cm
Rhombus ki chaaron sides barabar. Perimeter = chaaron ka jod = 7 + 7 + 7 + 7 = 4 × 7 = 28 cm.
Q6. Regular polygon ke ek vertex pe interior angle + exterior angle = _____
Jawab aur kaaran
Jawab: 180°
Interior aur exterior angle ek hi seedhi line pe saath-saath bante hain (linear pair), aur seedhi line ka angle 180° hota hai.
Q7. Jis polygon ki saari sides aur saare angles barabar hon, use _____ polygon kehte hain.
Jawab aur kaaran
Jawab: regular
Jaise square (regular quadrilateral) aur equilateral triangle (regular triangle). Rhombus regular nahi, kyunki uske angles barabar zaroori nahi.
Q8. Trapezium PQRS mein PQ ∥ SR hai. Tirchhi side PS ke dono angles ∠P aur ∠S ka jod _____ hota hai.
Jawab aur kaaran
Jawab: 180°
PQ aur SR parallel hain, aur PS dono ko kaatti hai. Aisi kaatne wali line ke ek hi taraf, parallel lines ke beech ke do angles ka jod 180° hota hai. To ∠P + ∠S = 180°.
— 📌 charumam.com
✅❌ Section C: True / False
Batao statement sahi (True) hai ya galat (False). Galat ho to use sahi karke likho.
Q1. Har rectangle ek parallelogram hai.
Jawab aur kaaran
✅ True
Sahi. Rectangle mein dono jodi saamne ki sides parallel hoti hain, aur parallelogram ki pehchaan yahi hai. (Rectangle = aisa parallelogram jiske saare angles 90°.)
Q2. Har rhombus ek square hai.
Jawab aur kaaran
❌ False
Galat. Rhombus ki chaaron sides barabar hoti hain par angles 90° zaroori nahi (jaise 60°, 120°, 60°, 120° wala rhombus). Sahi: “Har square ek rhombus hai, par har rhombus square nahi.”
Q3. Kisi quadrilateral ke chaar angles 100°, 100°, 100° aur 80° ho sakte hain.
Jawab aur kaaran
❌ False
Galat. Jod: 100 + 100 + 100 + 80 = 380, jo 360 nahi hai. Quadrilateral ke angles ka jod hamesha 360° hota hai, isliye aisa quadrilateral ban hi nahi sakta.
Q4. Square ke diagonals barabar bhi hote hain aur ek-doosre pe 90° bhi.
Jawab aur kaaran
✅ True
Sahi. Square rectangle bhi hai (isliye diagonals barabar) aur rhombus bhi (isliye diagonals 90° pe).
Q5. Sirf chaaron sides ki lambaai de do, to ek hi (unique) quadrilateral banta hai.
Jawab aur kaaran
❌ False
Galat. Chaar sides se shape “hil” sakti hai: jaise 5 cm ki chaar dandiyon se square bhi ban sakta hai aur dabaane pe tirchha rhombus bhi. Unique banane ke liye 5 naap chahiye (jaise 4 sides + 1 diagonal).
Q6. Kisi regular polygon ka har exterior angle 50° ho sakta hai.
Jawab aur kaaran
❌ False
Galat. Sides = 360 ÷ 50 = 7.2 (7.2 = 7 aur thoda zyada, ek point wala number). Sides ki ginti poora number (1, 2, 3 …) honi chahiye; 7.2 sides nahi ho sakti. Sahi: exterior angle aisa hona chahiye jo 360 ko poora-poora baant de (jaise 40°, 45°, 60°, 72°).
Q7. Parallelogram ke diagonals ek-doosre ko bisect (aadha-aadha) karte hain.
Jawab aur kaaran
✅ True
Sahi. Jahan diagonals milte hain, woh point dono diagonals ka beech ka point hota hai. Jaise AC = 10 cm ho to O se A tak 5 cm aur O se C tak 5 cm.
Q8. Triangle ke teeno angles ka jod 360° hota hai.
Jawab aur kaaran
❌ False
Galat. Triangle ke angles ka jod 180° hai. 360° quadrilateral ka jod hai (jo do triangles se banta hai: 2 × 180 = 360).
— 📌 charumam.com
🟢 Section D: Short Answer Questions
2 se 3 marks wale sawaal. Har step likhne ki aadat daalo.
Q1. Ek quadrilateral ke angles 3 : 4 : 5 : 6 ke ratio (anupaat) mein hain. Chaaron angles nikaalo.
Step-by-step hal
Step 1 (ratio ka matlab): 360° ko hisson mein baanta hai: pehle angle ko 3 hisse, doosre ko 4, teesre ko 5, chauthe ko 6. Sab hisse barabar size ke.
Step 2: kul hisse = 3 + 4 + 5 + 6 = 18.
Step 3: quadrilateral ke angles ka jod 360°, to ek hissa = 360 ÷ 18 = 20°.
Step 4: angles: 3 × 20 = 60°, 4 × 20 = 80°, 5 × 20 = 100°, 6 × 20 = 120°.
Check: 60 + 80 + 100 + 120 = 360 ✅
Q2. Ek regular polygon ka har interior angle 150° hai. Usme kitni sides hain?
Step-by-step hal
Step 1: pehle exterior angle nikaalo. Interior + exterior = 180° (ek seedhi line), to exterior = 180 − 150 = 30°.
Step 2: saare exterior angles ka jod 360°, aur regular polygon mein sab barabar. To sides = 360 ÷ 30 = 12.
Check: 12 sides ke interior angles ka jod = (12 − 2) × 180 = 10 × 180 = 1800, aur 1800 ÷ 12 = 150 ✅
Q3. Parallelogram PQRS mein ∠P = 70° hai. Baaki teeno angles nikaalo.
Step-by-step hal
Step 1: saamne ke angles barabar: ∠R (∠P ke saamne) = 70°.
Step 2: padosi angles ka jod 180°: ∠Q = 180 − 70 = 110°.
Step 3: ∠S, ∠Q ke saamne hai, to ∠S = 110°.
Check: 70 + 110 + 70 + 110 = 360 ✅
Q4. Kite ABCD mein AB = AD = 5 cm aur CB = CD = 8 cm hai. (a) Perimeter nikaalo. (b) Agar ∠B = 100° ho, to ∠D kitna hoga? Kaaran do.
Step-by-step hal
(a) perimeter = saari sides ka jod = 5 + 5 + 8 + 8 = 26 cm.
(b) Step 1: vertex B pe do alag lambaai wali sides milti hain (AB = 5 cm aur BC = 8 cm). Vertex D pe bhi (AD = 5 cm aur DC = 8 cm).
Step 2: kite ko diagonal AC pe modo: AB, AD pe gir jaati hai aur CB, CD pe (kyunki woh barabar hain). Isliye ∠B, ∠D pe bilkul fit baithta hai: kite mein alag-alag sides ke beech wale do angles barabar hote hain.
Jawab: ∠D = 100°.
Q5. Hexagon (6 sides) ke (a) interior angles ka jod aur (b) diagonals ki ginti nikaalo.
Step-by-step hal
(a) Step 1: ek vertex se saare diagonals kheecho: hexagon 6 − 2 = 4 triangles mein bantta hai.
Step 2: jod = 4 × 180 = 720°.
(b) Step 1: har vertex se 6 − 3 = 3 diagonals (khud aur do padosi chhod kar).
Step 2: 6 vertices × 3 = 18, par har diagonal do baar gina, to 18 ÷ 2 = 9.
Q6. Kya in naapon se quadrilateral ABCD unique banega? (a) AB = 4 cm, BC = 5 cm, CD = 6 cm, DA = 5 cm (b) AB = 4 cm, BC = 5 cm, CD = 6 cm, DA = 5 cm, AC = 7 cm
Step-by-step hal
(a) sirf 4 naap (chaar sides) hain. Shape hil sakti hai, to unique nahi; alag-alag quadrilateral ban jaayenge.
(b) 5 naap (4 sides + diagonal AC). Diagonal AC shape ko do triangles mein baant deta hai, to jaanch karo ki dono triangle ban sakte hain (kisi bhi do sides ka jod teesri se bada):
Triangle ABC (4, 5, 7): 4 + 5 = 9, jo 7 se bada ✅ (4 + 7 = 11 > 5 aur 5 + 7 = 12 > 4 bhi ✅).
Triangle ACD (7, 6, 5): 6 + 5 = 11, jo 7 se bada ✅ (baaki jod bhi bade ✅).
Jawab: (b) se unique quadrilateral banega.
— 📌 charumam.com
📘 Section E: Long Answer Questions
4 se 5 marks wale sawaal. Hal mein har step ka kaaran bhi likha hai.
Q1. Dikhao ki kisi bhi quadrilateral ke angles ka jod 360° hota hai. Phir isi tareeke se pentagon, hexagon aur octagon ke interior angles ka jod nikaalo aur ek table banao.
Step-by-step hal
Step 1 (quadrilateral): quadrilateral ABCD mein diagonal AC kheecho. Do triangle bane: ABC aur ACD.
Step 2: triangle ABC ke angles ka jod 180°, triangle ACD ka bhi 180°. Vertex A ka angle do hisson mein bata hai (ek hissa har triangle mein), aur vertex C ka bhi. Isliye dono triangles ke saare angles milkar ∠A + ∠B + ∠C + ∠D hi banate hain.
Step 3: ∠A + ∠B + ∠C + ∠D = 180 + 180 = 360° ✅
Step 4 (baaki polygons): ek hi vertex se saare diagonals kheecho. Har baar (sides − 2) triangle bante hain:
| Polygon | Sides | Triangle | Interior angles ka jod |
|---|---|---|---|
| Quadrilateral | 4 | 2 | 2 × 180 = 360° |
| Pentagon | 5 | 3 | 3 × 180 = 540° |
| Hexagon | 6 | 4 | 4 × 180 = 720° |
| Octagon | 8 | 6 | 6 × 180 = 1080° |
Pattern: jod = (sides − 2) × 180°.
Q2. Quadrilateral PQRS banao (construction) jisme PQ = 4.5 cm, QR = 5.5 cm, RS = 4 cm, SP = 6 cm aur diagonal PR = 7 cm hai. Har step likho aur batao ki yeh banana mumkin kyun hai.
Step-by-step hal
Pehle jaanch (kya ban sakta hai?): diagonal PR shape ko do triangles mein baantega: PQR (4.5, 5.5, 7) aur PRS (7, 4, 6).
Triangle PQR: 4.5 + 5.5 = 10, jo 7 se bada ✅. Triangle PRS: 4 + 6 = 10, jo 7 se bada ✅. To dono triangle banenge.
Step 1: ruler se 7 cm ki line PR kheecho (yeh diagonal hai, pehle isi ko banate hain).
Step 2: compass ko 4.5 cm khol kar, nok P pe rakho aur PR ke neeche ek arc (chaap) lagao.
Step 3: compass ko 5.5 cm khol kar, nok R pe rakho aur neeche ek arc lagao jo pichhle arc ko kaate. Kaatne wala point Q hai. (Q, P se 4.5 cm aur R se 5.5 cm door hai, kyunki dono arc usi doori pe hain.)
Step 4: compass 6 cm khol kar, P se PR ke upar arc lagao.
Step 5: compass 4 cm khol kar, R se upar arc lagao jo pichhle arc ko kaate. Yeh point S hai.
Step 6: PQ, QR, RS aur SP ko ruler se jodo. PQRS ready.
Kyun upar-neeche? Q aur S diagonal ke alag-alag taraf hone chahiye, warna shape convex quadrilateral nahi banegi.
Check: ruler se PQ, QR, RS, SP naap kar dekho: 4.5, 5.5, 4 aur 6 cm aane chahiye.
Q3. Parallelogram ABCD mein ∠A = (3x + 10)° aur ∠C = (5x − 30)° hai. x ki value aur chaaron angles nikaalo.
Step-by-step hal
Step 1 (kaun sa niyam?): ∠A aur ∠C saamne ke angles hain, aur parallelogram mein saamne ke angles barabar. To 3x + 10 = 5x − 30. (3x matlab 3 guna x.)
Step 2: x wale terms (equation ke tukde jo + ya − se alag hote hain, jaise 3x, 10, 5x, 30) ek taraf laane ke liye dono taraf se 3x ghatao (taraazu ke dono palde se same cheez hatao to barabari bani rehti hai): 10 = 2x − 30.
Step 3: −30 hataane ke liye dono taraf 30 jodo: 10 + 30 = 40, to 40 = 2x.
Step 4: 2 guna x = 40, to dono taraf 2 se bhaag: x = 40 ÷ 2 = 20.
Step 5 (angles): ∠A = 3 × 20 + 10 = 60 + 10 = 70°. ∠C = 5 × 20 − 30 = 100 − 30 = 70° (barabar ✅).
Step 6: padosi angles ka jod 180°: ∠B = 180 − 70 = 110°, aur ∠D (∠B ke saamne) = 110°.
Check: 70 + 110 + 70 + 110 = 360 ✅
Q4. Ek pentagon ke chaar exterior angles 70°, 80°, 60° aur 90° hain. (a) Paanchva exterior angle nikaalo. (b) Paanchon interior angles nikaalo aur check karo ki unka jod sahi hai.
Step-by-step hal
(a) Step 1: saare exterior angles ka jod 360°.
Step 2: chaaron ka jod: 70 + 80 + 60 + 90 = 300.
Step 3: paanchva = 360 − 300 = 60°.
(b) Step 1: har vertex pe interior = 180° − exterior (linear pair):
180 − 70 = 110°, 180 − 80 = 100°, 180 − 60 = 120°, 180 − 90 = 90°, aur paanchve ke liye 180 − 60 = 120°.
Step 2 (check): 110 + 100 + 120 + 90 + 120 = 540. Pentagon ke interior angles ka jod (5 − 2) × 180 = 540 hona chahiye ✅
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🧩 Section F: HOTS (Soch Wale Sawaal)
HOTS matlab Higher Order Thinking Skills: aise sawaal jinme formula se aage thoda dimaag lagana padta hai.
Q1. Kya kisi regular polygon ka har interior angle 100° ho sakta hai? Kaaran do.
Soch aur hal dekho
Soch: interior se exterior nikaalo, phir dekho sides poora number aata hai ya nahi.
Exterior = 180 − 100 = 80°. Sides = 360 ÷ 80 = 4.5.
Sides 4.5 nahi ho sakti (aadhi side nahi hoti). Jawab: nahi.
Q2. Farsh pe ek hi tarah ki regular tiles bina gap ke bichhaani hain. Square aur regular hexagon tiles se ho jaata hai, par regular pentagon se nahi. Kyun?
Soch aur hal dekho
Soch: jis point pe tiles ke kone milte hain, wahan ka poora chakkar 360° hai. Kono ka jod bilkul 360° banna chahiye.
Square: har angle 90°. 4 kone milein: 4 × 90 = 360 ✅
Regular hexagon: har angle 120°. 3 kone: 3 × 120 = 360 ✅
Regular pentagon: har angle = 540 ÷ 5 = 108°. 3 kone: 3 × 108 = 324 (36° ka gap bacha). 4 kone: 4 × 108 = 432 (360 se zyada, tiles chadh jaayengi). 108 se 360 poora nahi bantta, isliye gap rahega.
Q3. Rahul kehta hai: “Mere quadrilateral ke teen angles 90° hain, to yeh pakka rectangle hai.” Kya woh sahi hai?
Soch aur hal dekho
Step 1: chautha angle = 360 − (90 + 90 + 90) = 360 − 270 = 90°.
Step 2: to chaaron angles 90° hain. Rectangle ki pehchaan hi yahi hai: chaaron angles 90° wala quadrilateral.
Jawab: haan, Rahul sahi hai (yeh square bhi ho sakta hai, jo ek khaas rectangle hi hai).
Q4. Ek concave quadrilateral (teer ke sire jaisa, ek kona andar dhansa) ke angles 30°, 40° aur 30° hain, aur chautha andar dhansa reflex angle hai. Chautha angle kitna hai? Kya yahan bhi jod 360° chalega?
Soch aur hal dekho
Soch: andar dhanse (reflex) kone wale vertex se doosre kone tak diagonal kheecho; yeh diagonal shape ke andar rehta hai aur use do triangles mein baant deta hai. To yahan bhi jod 180 + 180 = 360°.
Hisaab: chautha angle = 360 − (30 + 40 + 30) = 360 − 100 = 260°.
260° reflex hai (180° se bada) ✅ jo concave shape se mel khaata hai.
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🔑 Answer Key (Ek Nazar Mein)
Pehle saare sawaal khud karo, phir yahan milaao. Jahan galti ho, upar us sawaal ka step-by-step hal dobara padho.
| Section | Answers |
|---|---|
| A. MCQ | 1: (c) · 2: (d) · 3: (b) · 4: (b) · 5: (b) · 6: (b) · 7: (c) · 8: (b) · 9: (c) · 10: (c) |
| B. Fill in the Blanks | 1: 6 · 2: 360° · 3: barabar · 4: 90 · 5: 28 cm · 6: 180° · 7: regular · 8: 180° |
| C. True / False | 1: T · 2: F · 3: F · 4: T · 5: F · 6: F · 7: T · 8: F |
| D. Short Answer | 1: 60°, 80°, 100°, 120° · 2: 12 sides · 3: ∠Q = 110°, ∠R = 70°, ∠S = 110° · 4: (a) 26 cm (b) ∠D = 100° · 5: (a) 720° (b) 9 · 6: (a) Nahi (b) Haan |
| E. Long Answer | 1: 360°; 540°, 720°, 1080° · 2: Construction: pehle PR = 7 cm, phir arcs se Q (neeche) aur S (upar) · 3: x = 20; ∠A = ∠C = 70°, ∠B = ∠D = 110° · 4: (a) 60° (b) 110°, 100°, 120°, 90°, 120° (jod 540°) |
| F. HOTS | 1: Nahi (sides 4.5 aati hain) · 2: 108° se 360° poora nahi bantta · 3: Haan, sahi · 4: 260°; haan, jod 360° hi |
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👩🏫 CharuMam Ka Tip
Geometry ke sawaal mein pehle chitra (rough figure) banao aur jo pata hai uspe likh do. Phir socho kaunsa niyam lagega: angles ka jod (triangle 180°, quadrilateral 360°, polygon (n − 2) × 180°), exterior ka jod 360°, ya khaas shape ki property (saamne ke angles barabar, padosi ka jod 180°, diagonals aadhe). Aakhir mein check line zaroor likho: saare angles jodo aur dekho jod sahi aata hai. 🔷
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🤖 Aur Practice: AI Tutor Se Poochho
Neeche wali line copy karke CharuMam AI Tutor (tutor.charumam.com) mein paste karo. Tutor naye sawaal dega aur galti pe samjhayega:
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👉 Aage Kya?
Pehle chapter Quadrilaterals ke posts padh lo (link neeche): wahan rectangle, square, parallelogram, rhombus, kite aur trapezium ki har property tark ke saath samjhaayi hai. Phir yeh practice dobara karo aur score milao. Agla Exemplar unit: Playing with Numbers (Number Play chapter ke saath), jald aa raha hai!
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