which of the following is correct?

Question

Comments

Correct answer is :- (1) ∠TRQ + ∠QSR  

Explanation:- 

Since, ∆PTR ≅ ∆QTS, Therefore 

PT = QT, TR = TS, PR = QS, ∠PTR = ∠QTS, ∠TRP = ∠TSQ  and ∠RPT = ∠SQT

Let, a = ∠TRQ,  b = ∠QSR and x = ∠PTQ 

Thus, 

In ∆TPQ 

∠TPQ = ∠TQP = 90 -      [Angle Sum and Isosceles Triangle]

Similarly, In ∆TRS, 

∠TRS = ∠TSR = 90 -       

Now, a = 90 -  - y [Angle sum in quadrilateral]

Also, 90 -  - y + b = 90 

y = b 

∠TQS = 90 -  

= 90 -  - y + y        [Adding and subtracting 'y']

= ∠TRQ + ∠QSR               [∠TRQ = a and∠QSR = b] 

Therefore Proved