Class 7 | Ganita Prakash Part-II | Chapter 1
🔀 Geometric Twins
Congruence · SSS · SAS · ASA · Isosceles Triangles
🔀 1.1 — What are Congruent Figures?
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📖 The Signboard Story
A symbol on a signboard needs to be exactly recreated on another board.
To copy it perfectly, we need the right measurements!
A symbol on a signboard needs to be exactly recreated on another board.
To copy it perfectly, we need the right measurements!
Congruent Figures = Exact Copies! 👯
Two figures are congruent if they have the same shape AND same size.
Two figures are congruent if they have the same shape AND same size.
- They can be rotated or flipped and still be congruent
- One can be superimposed exactly over the other
- Symbol: △ABC ≅ △DEF
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🔑 Key Insight!
To reconstruct a triangle, you only need 3 specific measurements.
Just AB and BC (two sides) are NOT enough — you also need the angle between them (∠ABC)!
Many different triangles can have the same two sides but different shapes.
To reconstruct a triangle, you only need 3 specific measurements.
Just AB and BC (two sides) are NOT enough — you also need the angle between them (∠ABC)!
Many different triangles can have the same two sides but different shapes.
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📐 Congruence Basics
Corresponding Parts:
When △ABC ≅ △PQR:
When △ABC ≅ △PQR:
- A ↔ P, B ↔ Q, C ↔ R (vertices match)
- AB = PQ, BC = QR, CA = RP (sides match)
- ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R (angles match)
CPCT Rule:
Corresponding Parts of Congruent Triangles are equal.
If triangles are proved congruent → all corresponding sides & angles are equal!
Corresponding Parts of Congruent Triangles are equal.
If triangles are proved congruent → all corresponding sides & angles are equal!
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🔺 1.2 — Triangle Congruence Rules
| Rule | Full Name | What you need |
|---|---|---|
| SSS | Side-Side-Side | All 3 sides equal |
| SAS | Side-Angle-Side | 2 sides + included angle |
| ASA | Angle-Side-Angle | 2 angles + included side |
| RHS | Right-Hypotenuse-Side | Right angle + hypotenuse + one side |
Understanding Each Rule:
🔷 SSS: AB=PQ, BC=QR, CA=RP → △ABC ≅ △PQR ✅
🔷 SAS: AB=PQ, ∠B=∠Q, BC=QR → △ABC ≅ △PQR ✅ (angle must be BETWEEN the two sides!)
🔷 ASA: ∠B=∠Q, BC=QR, ∠C=∠R → △ABC ≅ △PQR ✅ (side must be BETWEEN the two angles!)
🔷 RHS: Right angle + hyp. + one side equal → ✅
🔷 SSS: AB=PQ, BC=QR, CA=RP → △ABC ≅ △PQR ✅
🔷 SAS: AB=PQ, ∠B=∠Q, BC=QR → △ABC ≅ △PQR ✅ (angle must be BETWEEN the two sides!)
🔷 ASA: ∠B=∠Q, BC=QR, ∠C=∠R → △ABC ≅ △PQR ✅ (side must be BETWEEN the two angles!)
🔷 RHS: Right angle + hyp. + one side equal → ✅
⚠️ AAA is NOT a congruence rule!
Just because two triangles have the same angles doesn't mean they're congruent — they could be different sizes (similar, not congruent!)
e.g., a small and a big equilateral triangle both have angles 60°-60°-60° but are NOT congruent.
Just because two triangles have the same angles doesn't mean they're congruent — they could be different sizes (similar, not congruent!)
e.g., a small and a big equilateral triangle both have angles 60°-60°-60° but are NOT congruent.
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📏 1.3 — Isosceles & Equilateral Triangles
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Isosceles Triangle theorem:
If two sides of a triangle are equal → the angles opposite to them are equal!
AB = AC → ∠B = ∠C
Proof: Using congruence (SAS or SSS on the two halves)!
If two sides of a triangle are equal → the angles opposite to them are equal!
AB = AC → ∠B = ∠C
Proof: Using congruence (SAS or SSS on the two halves)!
Equilateral Triangle:
All 3 sides equal → All 3 angles equal = 60° each
It's a special isosceles triangle! (actually isosceles in three ways!)
All 3 sides equal → All 3 angles equal = 60° each
It's a special isosceles triangle! (actually isosceles in three ways!)
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📝 Quick Cheatsheet
Congruent = same
shape AND size! 👯
Can rotate/flip ✅
shape AND size! 👯
Can rotate/flip ✅
SSS ✅
SAS ✅
ASA ✅
AAA ❌
SAS ✅
ASA ✅
AAA ❌
CPCT:
Congruent parts
are all equal!
Congruent parts
are all equal!
Isosceles:
Equal sides →
equal angles!
Equal sides →
equal angles!
🌟 Chapter 1 Complete! 🌟
Made with ❤️ by @edugrown
Made with ❤️ by @edugrown