Given :

Δ ABC is an isosceles triangle and AB = AC.

AD is altitude

∴ ∠ADB = ∠ADC = 90°.

(i) In Δ BAD and Δ CAD,

⇒ ∠ADB = ∠ADC (Each equal to 90° as AD is altitude)

⇒ AB = AC (Given)

⇒ AD = AD (Common)

∴ Δ BAD ≅ Δ CAD (By R.H.S. Congruence rule)

We know that,

Corresponding parts of congruent triangles are equal.

∴ BD = CD (By C.P.C.T.)

Hence, proved that AD bisects BC.

(ii) Since, Δ BAD ≅ Δ CAD

∴ ∠BAD = ∠CAD (By C.P.C.T.)

Hence, proved that AD bisects ∠A.

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CBSE Class IX | Academic Year 2026-2027

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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)