In Fig. 6.43, if PQ ⊥ PS, PQ || SR, ∠ SQR = 28° and ∠ QRT = 65°, then find the values of x and y.

It is given in the question that:

PQ is perpendicular to PS

PQ parallel SR

∠SQR = 28^o

And,

∠QRT = 65^o

Now according to the question,

x + ∠SQR = ∠QRT (Alternate angles as QR is transversal)

x + 28^o = 65^o

x = 37^o

Also,

∠QSR = x

∠QSR = 37^o

Also,

∠QRS + ∠QRT = 180^o (Linear pair)

∠QRS + 65^o = 180^o

∠QRS = 115^o

Now,

∠P + ∠Q + ∠R + ∠S = 360^o (Sum of the angles in a quadrilateral)

90^o + 65⁰ + 115^o + ∠S = 360^o

270^o + y + ∠QSR = 360^o

270^o + y + 37^o = 360^o

307^o + y = 360^o

y = 53^o