In figure (1) given below, equilateral triangle EBC surmounts square square ABCD. Find angle BED represented by x.

△EBC is an equilateral triangle, so all sides are equal.

EB = BC = EC ……..(i)

In square all sides are equal

AD = CD = BC = AB ……..(ii)

From (i) and (ii) we get,

BC = EC = CD

⇒ EC = CD.

In △ECD,

EC = CD

⇒ ∠DEC = ∠CDE = a (let) (Angles opposite to equal sides are equal in isosceles triangle)

⇒ ∠C = ∠ECB + ∠BCD

∠ECB = 60° (As each angle of a equilateral triangle = 60°)

∠BCD = 90° (As each angle of a square = 90°)

⇒ ∠C = 60° + 90° = 150°.

⇒ ∠DEC + ∠CDE + ∠C = 180

⇒ a + a + 150° = 180°

⇒ 2a = 180° - 150°

⇒ 2a = 30°

⇒ a = 15°.

From figure,

x° = ∠BEC - ∠DEC = 60° - 15° = 45°.

Hence, x = 45.