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 = \(53^\circ\)