Rule : SSS - SAS - ASA - RHS
Grade 9 Mathematics : Chapter: Triangles
Congruence Rules:
SSS (Side–Side–Side) Congruence Rule
Rule: If the three sides of one triangle are equal to the three corresponding sides of another triangle, then the two triangles are congruent.
Note: No angles are needed to apply the SSS rule.
SSS (Side–Side–Side) āĻ¨āĻŋāĻ¯āĻŧāĻŽ : Rule : Click Here
āĻ¨āĻŋāĻ¯āĻŧāĻŽ :āĻ¯āĻĻāĻŋ āĻāĻāĻž āĻ¤ā§ā§°āĻŋāĻā§āĻā§° āĻ¤āĻŋāĻ¨āĻŋāĻāĻāĻž āĻŦāĻžāĻšā§ āĻāĻ¨ āĻāĻāĻž āĻ¤ā§ā§°āĻŋāĻā§āĻā§° āĻ¸āĻŽā§āĻŦāĻ¨ā§āĻ§āĻŋāĻ¤ āĻ¤āĻŋāĻ¨āĻŋāĻāĻāĻž āĻŦāĻžāĻšā§ā§° āĻ¸ā§āĻ¤ā§ āĻ¸āĻŽāĻžāĻ¨ āĻšāĻ¯āĻŧ, āĻ¤ā§āĻ¨ā§āĻ¤ā§ āĻ¤ā§ā§°āĻŋāĻā§āĻ āĻĻā§āĻāĻž āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯ (Congruent) āĻšāĻ¯āĻŧāĨ¤
āĻā§āĻāĻž : āĻā§āĻ¨ā§ āĻā§āĻŖā§° āĻĒā§ā§°āĻ¯āĻŧā§āĻāĻ¨ āĻ¨āĻžāĻ SSS āĻ¨āĻŋāĻ¯āĻŧāĻŽ āĻŦā§āĻ¯ā§ąāĻšāĻžā§° āĻā§°āĻŋāĻŦāĻ˛ā§āĨ¤
Exam Tip: āĻļā§āĻˇāĻ¤ āĻ˛āĻŋāĻāĻŋāĻŦ → ∴ âŗABC ≅ âŗPQR (by SSS rule)
AAS (Angle–Angle–Side) Congruence Rule : Click Here
Rule: Two triangles are congruent if any two pairs of corresponding angles and one pair of corresponding sides are equal.
Note: The side is NOT between the two angles
AAS (Angle–Angle–Side) āĻ¨āĻŋāĻ¯āĻŧāĻŽ
āĻ¨āĻŋāĻ¯āĻŧāĻŽ: āĻ¯āĻĻāĻŋ āĻĻā§āĻāĻž āĻ¤ā§ā§°āĻŋāĻā§āĻā§° āĻ¯āĻŋāĻā§āĻ¨ā§ āĻĻā§āĻāĻž āĻ¸āĻŽā§āĻŦāĻ¨ā§āĻ§āĻŋāĻ¤ āĻā§āĻŖ āĻā§°ā§ āĻāĻāĻž āĻ¸āĻŽā§āĻŦāĻ¨ā§āĻ§āĻŋāĻ¤ āĻŦāĻžāĻšā§ āĻ¸āĻŽāĻžāĻ¨ āĻšāĻ¯āĻŧ, āĻ¤ā§āĻ¨ā§āĻ¤ā§ āĻ¤ā§ā§°āĻŋāĻā§āĻ āĻĻā§āĻāĻž āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯ (Congruent) āĻšāĻ¯āĻŧāĨ¤
āĻā§āĻāĻž: āĻ¸ā§āĻ āĻŦāĻžāĻšā§āĻā§ āĻĻā§āĻāĻž āĻā§āĻŖā§° āĻŽāĻžāĻāĻ¤ āĻ¨āĻšāĻ¯āĻŧāĨ¤
Exam Tip: ASA āĻā§°ā§ AAS āĻĻā§āĻ¯āĻŧā§āĻāĻžāĻ āĻ¤ā§ā§°āĻŋāĻā§āĻ āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯ āĻĒā§ā§°āĻŽāĻžāĻŖ āĻā§°āĻŋāĻŦāĻ˛ā§ āĻŦā§āĻ¯ā§ąāĻšāĻžā§° āĻā§°āĻŋāĻŦ āĻĒāĻžā§°āĻŋ, āĻāĻŋāĻ¨ā§āĻ¤ā§ ASAāĻ¤ āĻŦāĻžāĻšā§ āĻŽāĻžāĻāĻ¤ āĻĨāĻžāĻā§, AASāĻ¤ āĻŦāĻžāĻšā§ āĻŽāĻžāĻāĻ¤ āĻ¨āĻžāĻĨāĻžāĻā§āĨ¤
ASA (Angle–Side–Angle) Congruence Rule : Click Here
Rule: Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of another triangle.
Note: The side must be between the two angles.
ASA (Angle–Side–Angle) āĻ¨āĻŋāĻ¯āĻŧāĻŽ
āĻ¨āĻŋāĻ¯āĻŧāĻŽ: āĻ¯āĻĻāĻŋ āĻāĻāĻž āĻ¤ā§ā§°āĻŋāĻā§āĻā§° āĻĻā§āĻāĻž āĻā§āĻŖ āĻā§°ā§ āĻ¸ā§āĻ āĻĻā§āĻāĻž āĻā§āĻŖā§° āĻŽāĻžāĻā§° āĻŦāĻžāĻšā§ āĻāĻ¨ āĻāĻāĻž āĻ¤ā§ā§°āĻŋāĻā§āĻā§° āĻ¸āĻŽā§āĻŦāĻ¨ā§āĻ§āĻŋāĻ¤ āĻĻā§āĻāĻž āĻā§āĻŖ āĻā§°ā§ āĻŽāĻžāĻā§° āĻŦāĻžāĻšā§ā§° āĻ¸ā§āĻ¤ā§ āĻ¸āĻŽāĻžāĻ¨ āĻšāĻ¯āĻŧ, āĻ¤ā§āĻ¨ā§āĻ¤ā§ āĻ¤ā§ā§°āĻŋāĻā§āĻ āĻĻā§āĻāĻž āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯ (Congruent) āĻšāĻ¯āĻŧāĨ¤
āĻā§āĻāĻž: āĻŦāĻžāĻšā§āĻā§ āĻĻā§āĻāĻž āĻā§āĻŖā§° āĻŽāĻžāĻāĻ¤ āĻĨāĻžāĻāĻŋāĻŦāĻ āĻ˛āĻžāĻāĻŋāĻŦāĨ¤
Exam Tip: āĻļā§āĻˇāĻ¤ āĻ˛āĻŋāĻāĻŋāĻŦ → ∴ âŗABC ≅ âŗPQR (by ASA rule)
SAS (Side–Angle–Side) Congruence Rule : Click Here
Rule: Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle.
Note: The angle must be between the two sides.
SAS (Side–Angle–Side) āĻ¨āĻŋāĻ¯āĻŧāĻŽ
āĻ¨āĻŋāĻ¯āĻŧāĻŽ : āĻ¯āĻĻāĻŋ āĻāĻāĻž āĻ¤ā§ā§°āĻŋāĻā§āĻā§° āĻĻā§āĻāĻž āĻŦāĻžāĻšā§ āĻā§°ā§ āĻ¸ā§āĻ āĻĻā§āĻāĻž āĻŦāĻžāĻšā§ā§° āĻŽāĻžāĻā§° āĻā§āĻŖ āĻāĻ¨ āĻāĻāĻž āĻ¤ā§ā§°āĻŋāĻā§āĻā§° āĻ¸āĻŽā§āĻŦāĻ¨ā§āĻ§āĻŋāĻ¤ āĻĻā§āĻāĻž āĻŦāĻžāĻšā§ āĻā§°ā§ āĻŽāĻžāĻā§° āĻā§āĻŖā§° āĻ¸ā§āĻ¤ā§ āĻ¸āĻŽāĻžāĻ¨ āĻšāĻ¯āĻŧ, āĻ¤ā§āĻ¨ā§āĻ¤ā§ āĻ¤ā§ā§°āĻŋāĻā§āĻ āĻĻā§āĻāĻž āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯ (Congruent) āĻšāĻ¯āĻŧāĨ¤
āĻā§āĻāĻž : āĻā§āĻŖāĻā§ āĻĻā§āĻāĻž āĻŦāĻžāĻšā§ā§° āĻŽāĻžāĻāĻ¤ āĻĨāĻžāĻāĻŋāĻŦāĻ āĻ˛āĻžāĻāĻŋāĻŦāĨ¤
Exam Tip: āĻļā§āĻˇāĻ¤ āĻ˛āĻŋāĻāĻŋāĻŦ → ∴ âŗABC ≅ âŗPQR (by SAS rule).
Summary: Valid vs. Invalid Congruence Rules : āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯ āĻ¨āĻŋāĻ¯āĻŧāĻŽ – āĻ¸āĻžā§°āĻžāĻāĻļ
Valid Rules (Use these!) | āĻŦā§āĻ§ āĻ¨āĻŋāĻ¯āĻŧāĻŽ (āĻŦā§āĻ¯ā§ąāĻšāĻžā§° āĻā§°āĻ)
- SAS (Side–Angle–Side) : āĻĻā§āĻāĻž āĻŦāĻžāĻšā§ āĻā§°ā§ āĻŽāĻžāĻā§° āĻā§āĻŖ āĻ¸āĻŽāĻžāĻ¨ → āĻ¤ā§ā§°āĻŋāĻā§āĻ āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯
- ASA (Angle–Side–Angle) : āĻĻā§āĻāĻž āĻā§āĻŖ āĻā§°ā§ āĻŽāĻžāĻā§° āĻŦāĻžāĻšā§ āĻ¸āĻŽāĻžāĻ¨ → āĻ¤ā§ā§°āĻŋāĻā§āĻ āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯
- AAS (Angle–Angle–Side) : āĻĻā§āĻāĻž āĻā§āĻŖ āĻā§°ā§ āĻā§āĻ¨ā§ āĻāĻāĻž āĻŦāĻžāĻšā§ āĻ¸āĻŽāĻžāĻ¨ → āĻ¤ā§ā§°āĻŋāĻā§āĻ āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯
- SSS (Side–Side–Side) : āĻ¤āĻŋāĻ¨āĻŋāĻāĻāĻž āĻŦāĻžāĻšā§ āĻ¸āĻŽāĻžāĻ¨ → āĻ¤ā§ā§°āĻŋāĻā§āĻ āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯
- RHS (Right–Hypotenuse–Side) : āĻ¸āĻŽāĻā§āĻŖā§ āĻ¤ā§ā§°āĻŋāĻā§āĻāĻ¤ āĻ āĻ§āĻŋāĻāĻŦāĻžāĻšā§ āĻā§°ā§ āĻāĻāĻž āĻŦāĻžāĻšā§ āĻ¸āĻŽāĻžāĻ¨ → āĻ¤ā§ā§°āĻŋāĻā§āĻ āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯
Invalid Rules (Don’t use!) / āĻ āĻŦā§āĻ§ āĻ¨āĻŋāĻ¯āĻŧāĻŽ (āĻŦā§āĻ¯ā§ąāĻšāĻžā§° āĻ¨āĻā§°āĻŋāĻŦ)
- AAA (Angle–Angle–Angle) : āĻŽāĻžāĻ¤ā§āĻ° āĻā§āĻŖāĻ¸āĻŽā§āĻš āĻ¸āĻŽāĻžāĻ¨ → āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯ āĻ¨āĻšāĻ¯āĻŧ, Similarity only
- SSA (Side–Side–Angle) : āĻĻā§āĻāĻž āĻŦāĻžāĻšā§ āĻā§°ā§ āĻā§āĻ¨ā§ āĻāĻāĻž āĻā§āĻŖ āĻ¸āĻŽāĻžāĻ¨ → Ambiguous case, āĻ¸āĻŽāĻĻā§āĻļā§āĻ¯ āĻ¨ā§āĻšā§ā§ąāĻž āĻĒāĻžā§°ā§
Exam Tip / āĻĒā§°ā§āĻā§āĻˇāĻžā§° āĻāĻŋāĻĒ: 1. Valid rules āĻ˛ā§ congruence proof āĻĻāĻŋāĻŦ. 2. Invalid rules āĻ˛āĻŋāĻāĻŋāĻ˛ā§ āĻ¨āĻŽā§āĻŦā§° āĻāĻāĻžāĻ āĻĻāĻŋāĻ¯āĻŧāĻž āĻš’āĻŦ āĻĒāĻžā§°ā§