Data: ∠XYZ = 64° and XY is produced upto P.

∠ZYP is bisected.

To Prove: ∠XYQ = ?

Reflex ∠QYP = ?

Proof: YQ bisects ∠ZYP

∴ Let ∠PYQ = ∠ZYQ = x°.

∠XYZ + ∠ZYQ + ∠QYP = 180°

(∵ ∠XYP is straight angle)

64 + x + x = 180

∴ 64 + 2x = 180

∴ 2x= 180 – 64

2x = 116

∴ x = \(\frac{116}{2}\) = 58°

∴ ∠PYQ = ∠ZYQ = 58°

(i) ∴ ∠XYQ= ∠XYZ + ∠ZYQ = 64 + 58

∠XYQ= 122°

(ii) Reflex ∠QYP = ∠PYX + ∠XYZ + ∠ZYQ

= 180 ° + 64° + 58°

= 180+122

∴ Reflex ∠QYP = 302°.