In the figure, if AC = AD = CD = BD; find angle ABC.

In △ ACD,

All sides are equal.

∴ ∠ADC = 60° (Each angle of equilateral triangle is 60°)

Since, CDB is a straight line.

∴ ∠ADC + ∠ADB = 180°

⇒ 60° + ∠ADB = 180°

⇒ ∠ADB = 180° - 60° = 120°.

In △ ADB,

⇒ AD = DB (Given)

∴ ∠DAB = ∠DBA = x (let) (Angles opposite to equal sides are equal)

By angle sum property of triangle,

⇒ ∠DAB + ∠DBA + ∠ADB = 180°

⇒ x + x + 120° = 180°

⇒ 2x = 180° - 120°

⇒ 2x = 60°

⇒ x = $\dfrac{60°}{2}$ = 30°.

⇒ ∠DBA = 30°.

From figure,

⇒ ∠ABC = ∠DBA = 30°.

Hence, ∠ABC = 30°.