Given :

⇒ AB = AC .......(1)

⇒ AD = AB ......(2)

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

⇒ AB = AC = AD

In an isosceles triangle ABC,

⇒ AB = AC

We know that,

Angles opposite to equal sides of a triangle are equal.

∴ ∠ACB = ∠ABC = x (let)

In Δ ACD,

⇒ AC = AD

We know that,

Angles opposite to equal sides of a triangle are equal.

∴ ∠ADC = ∠ACD = y (let)

From figure,

⇒ ∠BCD = ∠ACB + ∠ACD

⇒ ∠BCD = x + y .....(3)

In Δ BCD,

⇒ ∠ABC + ∠BCD + ∠ADC = 180° (Angle sum property of a triangle)

⇒ x + (x + y) + y = 180° [From equation (1), (2) and (3)]

⇒ 2(x + y) = 180°

Substituting value of (x + y) from equation (3) in above equation :

⇒ 2(∠BCD) = 180°

⇒ ∠BCD = $\dfrac{180°}{2}$

⇒ ∠BCD = 90°

Hence, proved that ∠BCD is a right angle.

BRIGHT TUTORIALS

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)