Class 7 | Ganita Prakash Part-II | Chapter 1
๐ Geometric Twins
Congruence ยท SSS ยท SAS ยท ASA ยท Isosceles Triangles
๐ 1.1 โ What are Congruent Figures?
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๐ 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! ๐ฏ
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: โณABC โ โณDEF
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๐ Key Insight!
To reconstruct a triangle, you only need 3 specific measurements.
Just AB and BC (two sides) are NOT enough โ you also need the angle between them (โ 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 โ you also need the angle between them (โ ABC)!
Many different triangles can have the same two sides but different shapes.
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๐ Congruence Basics
Corresponding Parts:
When โณABC โ โณPQR:
When โณABC โ โณPQR:
- A โ P, B โ Q, C โ R (vertices match)
- AB = PQ, BC = QR, CA = RP (sides match)
- โ A = โ P, โ B = โ Q, โ C = โ R (angles match)
CPCT Rule:
Corresponding Parts of Congruent Triangles are equal.
If triangles are proved congruent โ all corresponding sides & angles are equal!
Corresponding Parts of Congruent Triangles are equal.
If triangles are proved congruent โ all corresponding sides & angles are equal!
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๐บ 1.2 โ 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:
๐ท SSS: AB=PQ, BC=QR, CA=RP โ โณABC โ โณPQR โ
๐ท SAS: AB=PQ, โ B=โ Q, BC=QR โ โณABC โ โณPQR โ (angle must be BETWEEN the two sides!)
๐ท ASA: โ B=โ Q, BC=QR, โ C=โ R โ โณABC โ โณPQR โ (side must be BETWEEN the two angles!)
๐ท RHS: Right angle + hyp. + one side equal โ โ
๐ท SSS: AB=PQ, BC=QR, CA=RP โ โณABC โ โณPQR โ
๐ท SAS: AB=PQ, โ B=โ Q, BC=QR โ โณABC โ โณPQR โ (angle must be BETWEEN the two sides!)
๐ท ASA: โ B=โ Q, BC=QR, โ C=โ R โ โณABC โ โณPQR โ (side must be BETWEEN the two angles!)
๐ท RHS: Right angle + hyp. + one side equal โ โ
โ ๏ธ AAA is NOT a congruence rule!
Just because two triangles have the same angles doesn't mean they're congruent โ they could be different sizes (similar, not congruent!)
e.g., a small and a big equilateral triangle both have angles 60ยฐ-60ยฐ-60ยฐ but are NOT congruent.
Just because two triangles have the same angles doesn't mean they're congruent โ they could be different sizes (similar, not congruent!)
e.g., a small and a big equilateral triangle both have angles 60ยฐ-60ยฐ-60ยฐ but are NOT congruent.
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๐ 1.3 โ Isosceles & Equilateral Triangles
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Isosceles Triangle theorem:
If two sides of a triangle are equal โ the angles opposite to them are equal!
AB = AC โ โ B = โ C
Proof: Using congruence (SAS or SSS on the two halves)!
If two sides of a triangle are equal โ the angles opposite to them are equal!
AB = AC โ โ B = โ C
Proof: Using congruence (SAS or SSS on the two halves)!
Equilateral Triangle:
All 3 sides equal โ All 3 angles equal = 60ยฐ each
It's a special isosceles triangle! (actually isosceles in three ways!)
All 3 sides equal โ All 3 angles equal = 60ยฐ each
It's a special isosceles triangle! (actually isosceles in three ways!)
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๐ Quick Cheatsheet
Congruent = same
shape AND size! ๐ฏ
Can rotate/flip โ
shape AND size! ๐ฏ
Can rotate/flip โ
SSS โ
SAS โ
ASA โ
AAA โ
SAS โ
ASA โ
AAA โ
CPCT:
Congruent parts
are all equal!
Congruent parts
are all equal!
Isosceles:
Equal sides โ
equal angles!
Equal sides โ
equal angles!
๐ Chapter 1 Complete! ๐
Made with โค๏ธ by @edugrown
Made with โค๏ธ by @edugrown