In the figure (ii) given below, sides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E, the sides AD and BC are produced to meet at F. If x : y : z = 3 : 4 : 5, find the values of x, y and z.

In figure,

ABCD is a cyclic quadrilateral.

∠DAB = ∠BCE = x (Property of cyclic quadrilateral by which an exterior angle = opposite interior angle.)

In △BCE

∠CBE = 180° - (x° + y°)

From figure,

∠CBE and ∠CBA are linear pairs.

So,

⇒ ∠CBE + ∠CBA = 180°
⇒ 180° - (x° + y°) + ∠CBA = 180°
⇒ ∠CBA = 180° - 180° + (x° + y°)
⇒ ∠CBA = x° + y°

In △ABF,

So,

∠BAF + ∠ABF + ∠AFB = 180°
x + (x + y) + z = 180° (∵ From figure, ∠BAF = ∠DAB and ∠ABF = ∠CBA)

Given x : y : z = 3 : 4 : 5 , so x = 3k, y = 4k and z = 5k

3k + (3k + 4k) + 5k = 180°
15k = 180°
k = 12°

Hence, x = 3k = 3 × 12° = 36°, y = 4k = 4 × 12° = 48° and z = 5k = 5 × 12° = 60°.

Hence, the value of x = 36°, y = 48° and z = 60°.