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# In figure, if PQ is perpendicular to PS , PQ || SR , $$\angle{SQR}$$ = $$28^\circ$$ and $$\angle{QRT}$$ = $$65^\circ$$ , then find the values of x and y.

Here, as PQ || SR, by alternate angles axiom, we get,
$$\angle{PQR}$$ = $$\angle{QRT}$$

It is given that,
$$\angle{PQS}$$ = x, $$\angle{SQR}$$ = $$28^\circ$$ and $$\angle{QRT}$$ = $$65^\circ$$
$$\therefore$$ $$\angle{PQS}$$ + $$\angle{SQR}$$ = $$\angle{QRT}$$
$$\Rightarrow$$ x + $$28^\circ$$ = $$65^\circ$$
$$\Rightarrow$$ x = $$37^\circ$$ ...(i)

Now, considering right angled triangle PQS,
$$\angle{SPQ}$$ = $$90^\circ$$
$$\angle{SPQ}$$ + x + y = $$180^\circ$$ ....(Since Sum of all angles of a triangle is equal to 180)

$$\therefore$$ $$90^\circ$$ + $$37^\circ$$ + y = $$180^\circ$$
$$\Rightarrow$$ y = $$180^\circ$$ - $$127^\circ$$
$$\Rightarrow$$ y = $$53^\circ$$