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

In the figure,the sides $AB$ and $AC$ of $\Delta ABC$ are produced to points $E$ and $D$ respectively. If the bisectors $BO$ and $CO$ of $\angle CBE$ and $\angle BCD$ respectively meet at point $O$,then prove that $\angle BOC = 90^o - \frac{1}{2} \angle BAC$.

In the figure,if $AB \parallel CD$,$CD \parallel EF$ and $y: z = 3: 7$,find $x$. (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 $\angle ROS = \frac{1}{2}(\angle QOS - \angle POS)$.

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,if lines $PQ$ and $RS$ intersect at point $T$,such that $\angle PRT = 40^\circ$,$\angle RPT = 95^\circ$ and $\angle TSQ = 75^\circ$,find $\angle SQT$. (in $^\circ$)