Find the numerical value of x from the diagram given below.

In Δ ABC, such that AB = BC. And BC = BD. AB is extended till point E such that ∠EBD = 75°.

Let ∠BAC = x°.

If two sides of a triangle are equal, then two opposite angles are always equal.

∠BAC = ∠BCA = x° (∴ AB = BC)

Sum of all angles in triangle ABC is 180°.

⇒ ∠BAC + ∠BCA + ∠CBA = 180°

⇒ x° + x° + ∠CBA = 180°

⇒ 2x° + ∠CBA = 180°

⇒ ∠CBA = 180° - 2x°

Similarly, in Δ BDC,

Let ∠CBD = θ

If two sides of a triangle are equal, then two opposite angles are always equal.

∠CBD = ∠CDB = θ (∴ DB = BC)

Sum of all angles in triangle BDC is 180°.

⇒ ∠CBD + ∠CDB + ∠DCB = 180°

⇒ θ + θ + ∠DCB = 180°

⇒ 2θ + ∠DCB = 180°

⇒ ∠DCB = 180° - 2θ

∠DCB and ∠BCA form linear angle.

⇒ 180° - 2θ + x = 180°

⇒ - 2θ + x = 0

⇒ 2θ = x

∠EBD, ∠DBC and ∠CBA form linear angle.

⇒ ∠EBD + ∠DBC + ∠CBA = 180°

⇒ 75° + θ + 180° - 2x° = 180°

(∵ Substituting x = 2θ)

⇒ 75° + θ + 180° - 2(2θ) = 180°

⇒ 75° + θ - 4θ = 0

⇒ 75° - 3θ = 0

⇒ 3θ = 75°

⇒ θ = $\dfrac{75°}{3}$

⇒ θ = 25°

So, x = 2θ = 2 x 25° = 50°

Hence, the value of x = 50°.