In the figure (1) given below, AB = AD, BC = DC. Find ∠ABC.

Join BD.

In △ABD,

AB = AD

∴ ∠ABD = ∠ADB = a.

∠ABD + ∠ADB + ∠DAB = 180°

a + a + 54 = 180°

2a + 54° = 180°

2a = 126°

a = 63°.

In △BCD,

BC = CD

∴ ∠CDB = ∠DBC = b.

∠CDB + ∠DBC + ∠BCD = 180°

b + b + 116° = 180°

2b + 116° = 180°

2b = 64°

b = 32°.

From figure,

∠ABC = a + b = 63 + 32 = 95°.

Hence, ∠ABC = 95°.