Calculate the angles x, y and z if :

$\dfrac{x}{3} = \dfrac{y}{4} = \dfrac{z}{5}$

Let $\dfrac{x}{3} = \dfrac{y}{4} = \dfrac{z}{5}$ = k.

∴ x = 3k, y = 4k and z = 5k.

From figure,

∠BCP = ∠DCQ [Vertically opposite angles are equal.]

Exterior angle of a triangle is equal to the sum of two opposite interior angles.

∠ABC = ∠BCP + ∠BPC = x + y = 3k + 4k = 7k

∠ADC = ∠DCQ + ∠DQC = x + z = 3k + 5k = 8k.

ABCD is a cyclic quadrilateral.

We know that,

⇒ ∠ABC + ∠ADC = 180° [Sum of opposite angles in a cyclic quadrilateral = 180°]

⇒ 8k + 7k = 180°

⇒ 15k = 180°

⇒ k = $\dfrac{180}{15}$ = 12°.

x = 3k = 3 x 12° = 36°

y = 4k = 4 x 12° = 48°

z = 5k = 5 x 12° = 60°.

Hence, x = 36°, y = 48° and z = 60°.