Given :

AB = AC

OB is the bisectors of ∠B

⇒ ∠ABO = ∠OBC = $\dfrac{1}{2}∠B$.

OC is the bisectors of ∠C

⇒ ∠ACO = ∠OCB = $\dfrac{1}{2}∠C$.

(i) It is given that in triangle ABC, AB = AC

⇒ ∠ACB = ∠ABC

Dividing both sides of equation by 2, we get :

$\Rightarrow \dfrac{∠ACB}{2} = \dfrac{∠ABC}{2}$

⇒ ∠OCB = ∠OBC

We know that,

Sides opposite to equal angles of a triangle are also equal.

⇒ OB = OC.

Hence, proved that OB = OC.

(ii) In Δ OAB and Δ OAC,

⇒ AO = AO (Common)

⇒ OB = OC (Proved above)

⇒ AB = AC (Proved above)

∴ Δ OAB ≅ Δ OAC (By S.S.S. congruence rule)

We know that,

Corresponding parts of congruent triangles are equal.

⇒ ∠BAO = ∠CAO (By C.P.C.T.)

∴ AO bisects ∠A

Hence, proved that AO bisects ∠A.

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)