Given :

AB = PQ, BC = QR = x (let) and AM = PN.

(i) Given,

AM is the median of △ ABC

∴ BM = CM = $\dfrac{1}{2}BC = \dfrac{x}{2}$ .....(1)

Also, PN is the median of △ PQR

∴ QN = RN = $\dfrac{1}{2}QR = \dfrac{x}{2}$ ......(2)

From equation (1) and (2), we get :

BM = QN ..........(3)

Now, in △ ABM and △ PQN we have,

⇒ AB = PQ (Given)

⇒ BM = QN [From equation (3)]

⇒ AM = PN (Given)

∴ △ ABM ≅ △ PQN (By S.S.S. congruence rule)

Hence, proved that △ ABM ≅ △ PQN.

(ii) Since,

△ ABM ≅ △ PQN

We know that,

Corresponding parts of the congruent triangle are equal.

∠B = ∠Q (By C.P.C.T.) ...........(4)

Now, In △ ABC and △ PQR we have

⇒ AB = PQ (Given)

⇒ ∠B = ∠Q [From equation (4)]

⇒ BC = QR (Given)

∴ △ ABC ≅ △ PQR (By S.A.S. congruence rule)

Hence, proved that △ ABC ≅ △ PQR.

BRIGHT TUTORIALS

CBSE Class IX | Academic Year 2026-2027

9403781999

Excellence in Education

Triangles — Interactive Study Guide

Congruence Criteria at a Glance

Criterion	You Need	Remember
SAS	2 sides + included angle	Angle MUST be between the 2 sides
ASA	2 angles + included side	Side MUST be between the 2 angles
AAS	2 angles + any side	Side can be anywhere
SSS	3 sides	All three sides match
RHS	Right angle + hypotenuse + side	ONLY for right triangles

NEVER use AAA (only proves similarity) or SSA (ambiguous case)!

Proof Writing Pattern

Every congruence proof follows this pattern:

1. Identify the two triangles to compare.
2. List three pairs of equal elements (sides/angles) with reasons.
3. State the criterion (SAS/ASA/AAS/SSS/RHS).
4. Conclude congruence.
5. Use CPCT for any further deduction.

Quick Self-Check

1. In ΔABC, AB = AC. ∠B = 55°. Find ∠A. (∠C = 55°, ∠A = 70°)
2. Can a triangle have sides 2, 3, 6? (No: 2+3=5 < 6, violates triangle inequality)
3. In a triangle, the longest side is opposite to which angle? (The largest angle)