It is given that ∠XYZ = 64° and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects ∠ZYP, find ∠XYQ and reflex ∠QYP.

The figure is shown below:

In the given figure,

Line YQ bisect ∠ZYP.

∴ ∠QYP = ∠ZYQ = x (let) ………(1)

From figure,

⇒ ∠XYZ + ∠ZYQ + ∠QYP = 180° [Linear pairs]

⇒ 64° + x + x = 180° [From equation (1)]

⇒ 64° + 2x = 180°

⇒ 2x = 180° - 64°

⇒ 2x = 116°

⇒ x = $\dfrac{116°}{2}$

⇒ x = 58°.

⇒ ∠ZYQ = ∠QYP = x = 58°

Reflex ∠QYP = 360° - ∠QYP = 360° - 58° = 302°.

From figure,

⇒ ∠XYQ = ∠XYZ + ∠ZYQ

= 64° + 58°

= 122°.

Hence, ∠XYQ = 122° and Reflex ∠QYP = 302°.