In quadrilateral PQRS, the bisectors of ∠R and ∠S meet at point T (inside the quadrilateral) and ∠STR = 32.5°. If the ratio of ∠P to ∠Q is 4 : 9, then what is the difference between the measures of ∠Q and ∠P?​

Correct option is C

Given:

∠STR = 32.5^∘

Ratio of ∠P : ∠Q = 4 : 9

T is the point of intersection of angle bisectors of ∠R and ∠S

Solution: 

In $$\triangle STR$$ 

$$\frac12\angle PSR+\frac12\angle QRS +\angle STR = 180 \degree \\ \ \\ \frac12\angle PSR+\frac12\angle QRS = 180 \degree - 32.5 = 147.5\degree \\ \ \\ \implies \angle PSR+\angle QRS = 147.5\degree \times 2 = 295\degree$$ 

Now, In quadrilateral PQRS 

​$$\angle PSR + \angle QRS +\angle RQS +\angle QPS = 360\degree \\ \ \\ 295\degree + 4x+9x = 360\degree \\ \ \\ 13x = 360-295 = 65\degree \\ \ \\ x = 5\degree$$

Now,  ∠Q - ∠P  = 9x - 4x = 5x = $$5\times 5 = 25\degree$$​​​