What are the five criteria for congruence of triangles?

SSS (Side-Side-Side): all three pairs of corresponding sides equal. SAS (Side-Angle-Side): two sides and the included angle equal. ASA (Angle-Side-Angle): two angles and the included side equal. AAS (Angle-Angle-Side): two angles and a non-included side equal. RHS (Right angle-Hypotenuse-Side): for right triangles only. AAA and SSA are NOT valid congruence criteria.

Why is AAA not a valid congruence criterion?

AAA (all three angles equal) only proves similarity — it guarantees the triangles have the same shape but not the same size. For example, a triangle with angles 60°-60°-60° can have side lengths 1cm, 2cm, or 100cm — all similar but not congruent. To prove congruence, at least one pair of corresponding sides must be equal.

What is CPCT and how is it used in proofs?

CPCT (Corresponding Parts of Congruent Triangles are Equal) is a theorem stating that once two triangles are proved congruent, ALL their corresponding sides and angles are equal. Usage: First prove △ABD ≅ △ACD using SSS/SAS/ASA/AAS/RHS. Then state 'By CPCT, BD = DC' or any other corresponding parts you need.

Congruence of Triangles

Triangles • Class 9 Mathematics • NCERT • CBSE

Two triangles are congruent if they have exactly the same shape and size. The five congruence criteria are: SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side), and RHS (right angle-hypotenuse-side).

Key Formulas

SSS: 3 sides equal → congruent
SAS: 2 sides + included angle equal → congruent
ASA: 2 angles + included side equal → congruent
AAS: 2 angles + non-included side equal → congruent
RHS: right angle + hypotenuse + one side equal → congruent
CPCT: Corresponding Parts of Congruent Triangles are Equal

Frequently Asked Questions

What are the five criteria for congruence of triangles?: SSS (Side-Side-Side): all three pairs of corresponding sides equal. SAS (Side-Angle-Side): two sides and the included angle equal. ASA (Angle-Side-Angle): two angles and the included side equal. AAS (Angle-Angle-Side): two angles and a non-included side equal. RHS (Right angle-Hypotenuse-Side): for right triangles only. AAA and SSA are NOT valid congruence criteria.
Why is AAA not a valid congruence criterion?: AAA (all three angles equal) only proves similarity — it guarantees the triangles have the same shape but not the same size. For example, a triangle with angles 60°-60°-60° can have side lengths 1cm, 2cm, or 100cm — all similar but not congruent. To prove congruence, at least one pair of corresponding sides must be equal.
What is CPCT and how is it used in proofs?: CPCT (Corresponding Parts of Congruent Triangles are Equal) is a theorem stating that once two triangles are proved congruent, ALL their corresponding sides and angles are equal. Usage: First prove △ABD ≅ △ACD using SSS/SAS/ASA/AAS/RHS. Then state 'By CPCT, BD = DC' or any other corresponding parts you need.

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