Class 8 Maths · Chapter 4
Short answer:
Class 8 Maths Chapter 4 Quadrilaterals — parallelogram, rectangle, rhombus, square, trapezium aur kite ki properties, angle sum (360°), diagonal rules aur area formulas ka complete NCERT-style solution guide, step-by-step working ke saath.
Chapter 4 poore Class 8 Geometry ki neev hai — is chapter me hum quadrilaterals ki duniya explore karte hain: parallelogram, rectangle, rhombus, square, trapezium aur kite. In sabki properties, angle sum rule, diagonal behavior, aur area formulas aage chalkar mensuration aur coordinate geometry ke concepts ki base banate hain. Is guide me hum har type ke quadrilateral ki properties, angle sum rule, diagonal behavior, aur area formulas ko step-by-step solve karenge — bilkul CBSE marking scheme ki tarah, taaki exercise aur unit test dono me confidence rahe.
Chapter 4 Summary — 5 Minute Revision
Is chapter me humne quadrilaterals ke different types — parallelogram, rectangle, rhombus, square, trapezium, aur kite — ki properties detail me cover ki. Sabse important rule yaad rakhna hai: kisi bhi quadrilateral ke angles ka sum 360° hota hai. Diagonals ki properties har shape me alag hoti hain — parallelogram me bisect, rectangle me bisect + equal, rhombus me perpendicular bisect, aur square me sabhi teeno properties ek saath. Area formulas — parallelogram (base × height), rhombus/kite ((1/2) × d1 × d2), aur trapezium ((1/2) × sum of parallel sides × height) — exercises solve karne ke liye zaroori hain. Practice ke liye exercise questions aur in-text examples dobara solve karo taaki step-by-step working strong ho jaye.
In-Text Questions — Solutions
Ek quadrilateral ke teen angles 80°, 100°, aur 90° hain. Chautha angle nikaalo.
4th angle = 360° − (80° + 100° + 90°) = 360° − 270° = 90°
Answer: Chautha angle 90° hai.
Kya sabhi rectangles parallelogram hote hain? Kya sabhi parallelogram rectangle hote hain?
Haan, sabhi rectangles parallelogram hote hain kyunki unke opposite sides parallel aur equal hote hain. Nahi, sabhi parallelogram rectangle nahi hote — rectangle banne ke liye sabhi angles 90° hone chahiye, jo har parallelogram me nahi hota.
Ek rhombus ke sides ki length 10 cm hai aur ek diagonal 16 cm hai. Doosri diagonal nikaalo.
Rhombus ke diagonals perpendicular bisector hote hain, isliye half-diagonals aur side ek right triangle banate hain.
(half d1)² + (half d2)² = side²
8² + (half d2)² = 10²
64 + (half d2)² = 100 ⟹ (half d2)² = 36 ⟹ half d2 = 6
d2 = 12 cm
Exercise Questions — Solutions (Q1–Q6)
Ek quadrilateral ke angles 3x, 5x, 7x, aur 9x hain. In angles ki value nikaalo.
Quadrilateral ke sabhi angles ka sum 360° hota hai.
3x + 5x + 7x + 9x = 360°
24x = 360°
x = 15°
Toh angles honge: 3x = 45°, 5x = 75°, 7x = 105°, 9x = 135°.
Verification: 45 + 75 + 105 + 135 = 360° ✓
Ek parallelogram ABCD me angle A = 70° hai. Baaki teeno angles nikaalo.
Parallelogram me opposite angles equal hote hain aur adjacent angles supplementary (sum 180°) hote hain.
Angle A = Angle C = 70°
Angle B ek adjacent angle hai:
Angle A + Angle B = 180°
70° + Angle B = 180° ⟹ Angle B = 110°
Angle B = Angle D = 110°
Answer: Angle B = 110°, Angle C = 70°, Angle D = 110°.
Ek rhombus ke diagonals 16 cm aur 12 cm hain. Rhombus ka area nikaalo.
Rhombus ka area formula:
Area = (1/2) × d1 × d2
Area = (1/2) × 16 × 12
Area = (1/2) × 192 = 96 cm2
Answer: Rhombus ka area 96 cm² hai.
Ek trapezium ke parallel sides 10 cm aur 14 cm hain, aur unke beech ki distance (height) 6 cm hai. Trapezium ka area nikaalo.
Trapezium ka area formula:
Area = (1/2) × (sum of parallel sides) × height
Area = (1/2) × (10 + 14) × 6
Area = (1/2) × 24 × 6 = 72 cm2
Answer: Trapezium ka area 72 cm² hai.
Kya ek square ko rhombus kaha ja sakta hai? Kya ek rhombus ko square kaha ja sakta hai? Reason do.
Square ko rhombus kaha ja sakta hai — kyunki square ke sabhi sides equal hote hain aur opposite sides parallel hote hain, jo rhombus ki definition satisfy karta hai. Square ek special rhombus hai jisme sabhi angles 90° hote hain.
Rhombus ko square nahi kaha ja sakta — kyunki rhombus me sabhi angles equal (90°) hona zaroori nahi hai. Rhombus general case hai, square uska special case hai.
Ek kite ke diagonals nikaalo agar area 84 cm² ho aur ek diagonal 12 cm ho.
Kite ka area formula rhombus jaisa hi hota hai:
Area = (1/2) × d1 × d2
84 = (1/2) × 12 × d2
84 = 6 × d2
d2 = 14 cm
Answer: Doosri diagonal 14 cm hai.
Important Equations — Ek Nazar Me
| Property / Formula | Detail |
|---|---|
| Angle sum of quadrilateral | Sum of all interior angles = 360° |
| Sum of exterior angles | Sum of exterior angles (one at each vertex) = 360° |
| Parallelogram — opposite sides | Opposite sides equal aur parallel |
| Parallelogram — opposite angles | Opposite angles equal; adjacent angles supplementary (sum 180°) |
| Parallelogram — diagonals | Diagonals bisect each other |
| Rectangle | Parallelogram jisme har angle 90°; diagonals equal aur bisect karte hain |
| Rhombus | Parallelogram jisme sabhi sides equal; diagonals perpendicular bisector hote hain ek dusre ke |
| Square | Rectangle + Rhombus dono properties — sabhi sides equal, sabhi angles 90°, diagonals equal, perpendicular, bisect |
| Area of parallelogram | Area = base × height |
| Area of rhombus / kite | Area = (1/2) × d1 × d2 |
| Area of trapezium | Area = (1/2) × (sum of parallel sides) × height |
| Kite properties | Do pairs of adjacent equal sides; ek diagonal doosri ko perpendicular bisect karti hai |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Angle sum property ko galat apply karna — kai students triangle ka 180° wala rule quadrilateral pe bhi laga dete hain, jabki quadrilateral ka sum 360° hota hai.
- Rhombus aur square ke diagonals confuse karna — dono me diagonals perpendicular bisect karte hain, lekin sirf square me diagonals equal bhi hote hain, rhombus me nahi.
- Trapezium ka area formula bhool jana — height ko parallel sides ke beech ki perpendicular distance samajhne ke bajaye slant side maan lena galat hai.
- Parallelogram aur rectangle ki properties ko interchange kar dena — sabhi rectangle parallelogram hote hain, lekin sabhi parallelogram rectangle nahi hote.
- Kite ki diagonal properties me confusion — sirf ek diagonal doosri ko perpendicular bisect karti hai, dono nahi (jab tak special case na ho).
- Exterior angle sum ko galat samajhna — quadrilateral ke exterior angles ka sum bhi hamesha 360° hota hai, chahe sides ki sankhya kuch bhi ho, ye rule confuse ho jata hai.
Board-Style Important Questions
- School-level test aur unit tests me is chapter se angle-sum aur missing-angle wale questions frequently poochhe jaate hain.
- Rhombus aur kite ke diagonals se area nikaalne wale numerical problems practice ke liye common hain.
- Parallelogram ki properties (opposite sides/angles, diagonals bisect) prove karne wale short-answer questions important hain.
- Trapezium ka area formula apply karke real-life shape based questions (jaise field ya plot ka area) practice karna faydemand hai.
- Different quadrilaterals ko classify karna (Venn diagram ya property table ke through) conceptual questions me common hai.
Aksar Poochhe Jaane Wale Sawaal
Class 8 Maths Quadrilaterals chapter me kaunse topics cover hote hain?
Is chapter me quadrilaterals ki classification (parallelogram, rectangle, rhombus, square, trapezium, kite), unki properties, angle sum property (360°), diagonals ki properties, aur in figures ka area calculate karna cover hota hai.
Quadrilateral ke angles ka sum kitna hota hai aur ye kaise prove hota hai?
Kisi bhi quadrilateral ke angles ka sum 360° hota hai. Ise prove karne ke liye quadrilateral ko ek diagonal se do triangles me divide karte hain — har triangle ke angles ka sum 180° hota hai, toh dono milakar 360° ban jata hai.
Parallelogram, rhombus, rectangle aur square me kya difference hai?
Parallelogram: opposite sides parallel aur equal. Rectangle: parallelogram jisme sabhi angles 90°. Rhombus: parallelogram jisme sabhi sides equal. Square: rectangle + rhombus dono properties — sabhi sides equal aur sabhi angles 90°.
Kya square, rhombus aur rectangle dono ka special case hai?
Haan. Square me rectangle ki property (sabhi angles 90°) aur rhombus ki property (sabhi sides equal) dono ek saath hoti hain. Isliye square ko rectangle aur rhombus dono kaha ja sakta hai, lekin har rectangle ya har rhombus square nahi hota.
Trapezium aur parallelogram me kya farak hai?
Parallelogram me opposite sides ke dono pair parallel hote hain, jabki trapezium me sirf ek pair sides parallel hota hai. Isi wajah se dono ka area formula bhi alag hota hai — parallelogram me base × height, trapezium me (1/2) × (sum of parallel sides) × height.
Quadrilaterals ke exercise solve karte time sabse zyada mistakes kahan hoti hain?
Students aksar diagonal properties (jaise rhombus ke diagonals perpendicular bisect karte hain lekin rectangle ke nahi), aur area formulas (trapezium vs parallelogram) confuse kar dete hain. Angle sum property ko sahi se apply na karna bhi common mistake hai.
Class 8 Maths — Saare Chapters
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