It is given that $\angle XYZ = 64^o$ and $XY$ is produced to point $P$. If ray $YQ$ bisects $\angle ZYP$,find $\angle XYQ$ and reflex $\angle QYP$. (in $^o$)

In the figure,$POQ$ is a line. Ray $OR$ is perpendicular to line $PQ$. $OS$ is another ray lying between rays $OP$ and $OR$. Prove that $\angle ROS = \frac{1}{2}(\angle QOS - \angle POS)$.

In the figure,if $QT \perp PR$,$\angle TQR = 40^o$ and $\angle SPR = 30^o$,find $x$ and $y$.

In the figure,if $AB \parallel CD$,$CD \parallel EF$ and $y: z = 3: 7$,find $x$. (in $^o$)

In the figure,if $AB \parallel CD$,$\angle APQ = 50^o$ and $\angle PRD = 127^o$,find $x$ and $y$.

In the figure,the side $QR$ of $\Delta PQR$ is produced to a point $S$. If the bisectors of $\angle PQR$ and $\angle PRS$ meet at point $T$,then prove that $\angle QTR = \frac{1}{2} \angle QPR$.