In the figure (3) given below, calculate the size of each lettered angle.

From figure,

∠AEB = ∠DEC = 80° (Vertically opposite angles are equal).

In △AEB,

⇒ x + 54° + 80° = 180°

⇒ x + 134° = 180°

⇒ x = 46°.

AB = BC

⇒ ∠BAC = ∠ACB (As angles opposite to equal sides are equal.)

∴ ∠ACB = 54°

In △ABC,

∠ABC + ∠ACB + ∠BAC = 180°

⇒ (x + y)° + 54° + 54° = 180°

⇒ y° + 46° + 108° = 180°

⇒ y = 180° - 154° = 26°.

From figure,

∠ECD = ∠BAE = 54° (Alternate angles)

∠BCA + ∠ECD + z = 180°

54° + 54° + z = 180°

108° + z = 180°

z = 72°.

Hence, x = 46°, y = 26° and z = 72°.