Class 7 Maths · Chapter 9
Short answer: Congruence — do figures jinka shape aur size exactly same ho — ke concepts (line segments, angles, triangles, circles) aur SSS/SAS/ASA criteria ke saath NCERT Class 7 Maths Chapter 9 "Geometric Twins" ke saare exercise solutions, step-by-step CBSE marking scheme style.
Chapter 9 poore Class 7 Geometry ki neev hai jahan bachche sikhte hain ki do figures kab bilkul "twins" ki tarah same hoti hain — yani congruent. NCERT Class 7 Maths chapter wise solutions is series mein, "Geometric Twins" line segments, angles, triangles aur circles ki congruence ko step-by-step explain karta hai, saath hi SSS, SAS, aur ASA jaise congruence criteria bhi cover karta hai. Ye chapter purani book ke "Triangle and its Properties" jaise concepts ka ek naya, gehra treatment hai — jahan congruence ko sirf memorize nahi, balki logically prove karna sikhaya jaata hai. Aage chalkar ye foundation Class 8, 9, 10 ki geometry proofs (jaise circle theorems, similar triangles) ke liye critical ban jaati hai.
Is page par har exercise question ka detailed, marking-scheme-style solution diya gaya hai, saath hi common mistakes, in-text questions ke answers, aur practice questions bhi included hain — taaki concept crystal clear ho jaaye.
Chapter 9 Summary — 5 Minute Revision
Is chapter mein humne congruence ka concept detail mein cover kiya — do figures kab "geometric twins" kehlaati hain jab unka shape aur size exactly match kare. Line segments ke liye sirf length equal honi chahiye, angles ke liye measure equal, aur circles ke liye radius equal — lekin triangles ke liye humne SSS, SAS, aur ASA jaise formal criteria seekhe jinse bina figures ko physically overlap kiye bhi congruence prove kiya ja sakta hai.
Humne ye bhi dekha ki congruence aur similarity mein farak hai — AAA (sirf angles equal) se similarity prove hoti hai, congruence nahi. Har proof mein correct vertex correspondence aur step-wise reasoning likhna CBSE marking scheme ke liye critical hai.
Ye foundation Class 8, 9, aur 10 ki advanced geometry proofs, circle theorems, aur similar triangles jaise topics ke liye zaroori hai.
In-Text Questions — Solutions
Do figures congruent kab kehlaati hain?
Jab do figures ka shape aur size dono exactly same ho, aur ek figure ko doosri ke upar bina stretch/shrink kiye exactly rakha ja sake, tab unhe congruent kehte hain.
Example: do same-size ki photocopy sheets congruent hoti hain.
Kya sabhi squares congruent hote hain?
Nahi, sabhi squares similar toh hote hain (same shape), lekin congruent tabhi honge jab unki side length bhi equal ho.
Square 1 (side 3 cm) ≠ Square 2 (side 5 cm) — similar hain, congruent nahi
Do triangles mein sirf ek side aur ek angle equal ho, toh kya wo congruent honge?
Nahi, sirf ek side aur ek angle equal hone se congruence prove nahi hoti. Kam se kam 3 corresponding parts (jisme minimum ek side ho) match karne chahiye — jaise SSS, SAS, ya ASA criteria ke through.
Exercise Questions — Solutions (Q1–Q6)
Prove that two circles with the same radius are congruent using the SSS-equivalent circle rule.
Do do circles congruent kehlaate hain jab unki radius equal ho, kyunki tab har point centre se same distance par hoga, aur ek circle doosre par exactly superimpose ho jaayega.
Given: r₁ = r₂ = 5 cm
Step 1: Circle 1 ka har point centre O₁ se 5 cm door hai.
Step 2: Circle 2 ka har point centre O₂ se 5 cm door hai.
Step 3: Jab O₁ ko O₂ par rakha jaaye, har radius match karega — isliye circle 1 ≅ circle 2.
Hence proved, equal radius wale circles hamesha congruent hote hain.
In triangles ABC and PQR, AB = PQ, BC = QR, and AC = PR. State the congruence criterion and write the correspondence.
Yahan teeno sides equal hain, isliye SSS (Side-Side-Side) congruence rule lagega.
AB = PQ, BC = QR, AC = PR
Correspondence: A ↔ P, B ↔ Q, C ↔ R
∴ △ABC ≅ △PQR (by SSS rule)
Two line segments AB and CD are such that AB = 6 cm and CD = 6 cm. Are they congruent? Justify.
Haan, ye congruent hain.
AB = 6 cm, CD = 6 cm
Do line segments congruent hote hain jab unki length exactly barabar ho — orientation ya position matter nahi karti.
Chunki AB = CD = 6 cm, isliye AB ≅ CD.
In △DEF and △XYZ, DE = XY, ∠E = ∠Y, EF = YZ. Which congruence rule applies? Justify with reasoning.
Yahan do sides aur unke beech ka included angle diya hai, isliye SAS (Side-Angle-Side) rule apply hoga.
DE = XY, ∠E = ∠Y (included angle), EF = YZ
Step 1: ∠E, side DE aur EF ke beech mein hai — yahi included angle hai.
Step 2: Similarly ∠Y bhi XY aur YZ ke beech mein hai.
∴ △DEF ≅ △XYZ (by SAS rule)
Two squares have side lengths 4 cm and 4 cm respectively. Prove they are congruent, showing all corresponding parts match.
Ek square ke saare sides equal (4 cm) aur saare angles 90° hote hain.
Square 1: side = 4 cm, angles = 90° each
Square 2: side = 4 cm, angles = 90° each
Step 1: Sides ka correspondence: sabhi 4 sides equal (4 cm = 4 cm)
Step 2: Angles ka correspondence: sabhi angles equal (90° = 90°)
∴ Square 1 ≅ Square 2, kyunki har corresponding side aur angle match kar raha hai.
Two triangles have all three corresponding angles equal (∠A=∠P, ∠B=∠Q, ∠C=∠R) but sides are different. Are the triangles congruent? Explain with an example.
Nahi, ye triangles congruent nahi honge — sirf similar honge.
Example: △ABC ki sides 3, 4, 5 cm hain, aur △PQR ki sides 6, 8, 10 cm hain. Dono ke corresponding angles equal honge (kyunki ratio same hai, 1:2), lekin size different hai.
Congruence ke liye sirf angles equal hona kaafi nahi hai — AAA se sirf similarity prove hoti hai, congruence nahi. Isliye ye triangles congruent nahi, sirf similar hain.
Important Equations — Ek Nazar Me
| Concept | Rule / Condition |
|---|---|
| Congruent line segments | AB ≅ CD ⇔ AB = CD (length equal) |
| Congruent angles | ∠A ≅ ∠B ⇔ m∠A = m∠B (measure equal) |
| Congruent circles | Circle 1 ≅ Circle 2 ⇔ r₁ = r₂ |
| SSS Rule (Side-Side-Side) | AB=PQ, BC=QR, AC=PR ⇒ △ABC ≅ △PQR |
| SAS Rule (Side-Angle-Side) | AB=PQ, ∠B=∠Q (included), BC=QR ⇒ △ABC ≅ △PQR |
| ASA Rule (Angle-Side-Angle) | ∠A=∠P, AB=PQ (included side), ∠B=∠Q ⇒ △ABC ≅ △PQR |
| Corresponding parts of congruent triangles | △ABC ≅ △PQR ⇒ AB=PQ, BC=QR, AC=PR, ∠A=∠P, ∠B=∠Q, ∠C=∠R |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Students aksar congruence aur similarity ko confuse kar dete hain — sirf angles match karne par hi 'congruent' likh dete hain, jabki AAA sirf similarity prove karta hai, congruence nahi.
- Congruence statement likhte waqt vertices ka correspondence order galat likh dete hain (jaise △ABC ≅ △QPR likhna jab actual correspondence A↔P, B↔Q, C↔R ho) — is se poora answer galat ho jaata hai.
- SAS rule apply karte waqt 'included angle' ki jagah koi bhi random angle use kar lete hain, jabki angle strictly un do sides ke BEECH mein hi hona chahiye jinka mention ho raha hai.
- Diagram mein congruent parts dikhane ke liye tick marks (dashes) ka use nahi karte, jisse marking scheme mein steps clearly nazar nahi aate aur marks kat sakte hain.
- Circle ki congruence check karte waqt sirf 'dikhne mein same' bol dete hain, exact radius compare nahi karte — jabki congruence sirf measurement (r₁ = r₂) se hi prove hoti hai.
- Proof likhte waqt 'Given', 'To Prove', aur step-wise reasoning skip karke seedha final answer likh dete hain, jisse step marks miss ho jaate hain even agar final answer sahi ho.
Board-Style Important Questions
- Prove that two triangles are congruent if their corresponding sides are equal (SSS criterion), with a suitable example using actual measurements.
- Given two triangles with two sides and the included angle equal, show step-by-step that they are congruent by the SAS rule.
- Differentiate between congruent and similar figures with two real-life examples of each.
- Two circles have radii 7 cm and 7 cm respectively. State with reason whether they are congruent, and explain what would change if the radii were different.
- In a pair of triangles, ∠A=∠P, AB=PQ, ∠B=∠Q are given. Identify the congruence rule and complete the proof.
- Explain why AAA is not a valid congruence criterion for triangles, using a numerical counter-example.
Aksar Poochhe Jaane Wale Sawaal
Class 7 Maths chapter 9 'Geometric Twins' kis topic ke baare mein hai?
Ye chapter congruence ke baare mein hai — do figures jo shape aur size mein bilkul same hon unhe congruent kaha jaata hai. Chapter mein line segments, angles, triangles aur circles ki congruence cover hoti hai, saath hi SSS, SAS, ASA jaise congruence criteria bhi.
Congruence aur similarity mein kya farak hai?
Congruent figures ka shape AUR size dono same hote hain — ek doosre ke upar exactly fit ho jaate hain. Similar figures ka sirf shape same hota hai (same angles, proportional sides) lekin size different ho sakta hai. Har congruent figure similar bhi hoti hai, lekin har similar figure congruent nahi hoti.
Triangles ke liye kaunse congruence criteria NCERT Class 7 mein padhaye jaate hain?
Is chapter mein mukhya criteria hain: SSS (Side-Side-Side), SAS (Side-Angle-Side), aur ASA (Angle-Side-Angle). Inke through do triangles congruent hain ya nahi ye bina inko directly overlap kiye prove kiya ja sakta hai.
Kya sirf angles equal hone se do triangles congruent ho jaate hain?
Nahi. Agar sirf teeno angles equal hain (AAA) toh triangles sirf similar honge, congruent nahi — kyunki unki size (sides ki actual length) different ho sakti hai. Congruence prove karne ke liye kam se kam ek side ki length involve honi chahiye (jaise SSS, SAS, ya ASA).
Is chapter ki study exam ke liye kaise useful hai?
'Geometric Twins' ka concept aage Class 8, 9, 10 ki geometry (jaise triangle proofs, circle theorems) ki neev banata hai. NCERT Class 7 Maths chapter wise solutions practice karne se congruence criteria pakke ho jaate hain jo higher classes mein reasoning-based proofs ke liye zaroori hain.
Congruence symbol '≅' ka use kaise karte hain?
'≅' symbol congruence dikhane ke liye use hota hai, jaise △ABC ≅ △PQR ka matlab hai triangle ABC, triangle PQR ke congruent hai. Yahan vertices ka correspondence bhi important hota hai — A ↔ P, B ↔ Q, C ↔ R sahi order mein likhna zaroori hai, warna answer galat maana ja sakta hai.
Class 7 Maths — Saare Chapters
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