In the given figure, AP is perpendicular to BC, and AQ is the bisector of angle A. What will be the measure of angle PQA ?

From the given figure,

In $$\triangle\ $$ABC,

$$\angle\ $$A + $$\angle\ $$B + $$\angle\ $$C = $$180^{\circ\ }\ $$

$$=$$>  $$\angle\ $$A + $$60^{\circ}$$ + $$30^{\circ}$$ = $$180^{\circ\ }$$

$$=$$>  $$\angle\ $$A = $$180^{\circ}-90^{\circ}$$

$$=$$>  $$\angle\ $$A = $$90^{\circ}$$

AQ is the bisector of angle A

$$=$$>  $$\angle\ $$BAQ = $$\frac{90^{\circ}}{2}$$

$$=$$>  $$\angle\ $$BAQ = $$45^{\circ}$$

In $$\triangle\ $$ABQ,

$$\angle\ $$BAQ + $$\angle\ $$ABQ + $$\angle\ $$BQA = $$180^{\circ\ }\ $$

$$=$$>  $$45^{\circ}$$ + $$60^{\circ}$$ + $$\angle\ $$BQA = $$180^{\circ\ }$$

$$=$$> $$\angle\ $$BQA = $$180^{\circ}-105^{\circ}$$

$$=$$> $$\angle\ $$BQA = $$75^{\circ}$$

$$=$$> $$\angle\ $$PQA = $$\angle\ $$BQA = $$75^{\circ}$$

Hence, the correct answer is Option C