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Class 9 · CBSE / NCERT · Mathematics

Triangles — Class 9

Two triangles are congruent if they match exactly. Learn the five criteria that prove it without measuring every side and angle.

Congruent trianglesTriangles that are identical in shape and size — every corresponding side and angle matches.
SSS, SAS, ASA, AAS, RHSFive sufficient conditions to prove congruence without checking all six parts.
Isosceles triangle propertyAngles opposite equal sides are equal, and conversely.

Solved examples

Q1. In triangles ABC and PQR, AB = PQ, BC = QR and ∠B = ∠Q. Which congruence rule applies?

Two sides (AB, BC) and the INCLUDED angle (∠B, between them) are equal to the corresponding parts.

This matches the SAS (Side-Angle-Side) criterion.

So ΔABC ≅ ΔPQR by SAS.

Formula sheet

QuantityFormula
SSSall three sides equal
SAStwo sides & the included angle equal
ASAtwo angles & the included side equal
RHSright angle, hypotenuse & one side equal (right triangles only)

Common mistakes & exam wins

  • The angle in SAS must be the INCLUDED angle (between the two given sides) — not any angle.
  • RHS applies only to right-angled triangles.
  • AAA (all angles equal) does NOT prove congruence — it only proves similarity.
  • CPCT: "Corresponding Parts of Congruent Triangles" are equal — use it after proving congruence.

Frequently asked questions

What does CPCT mean?

"Corresponding Parts of Congruent Triangles (are equal)" — used to conclude that other sides/angles are equal once congruence is proved.

What is the RHS congruence rule?

If the hypotenuse and one side of a right triangle equal the hypotenuse and corresponding side of another right triangle, the triangles are congruent.

Does AAA prove two triangles are congruent?

No — AAA only proves the triangles are similar (same shape), not necessarily the same size.