In the figure,if $AB \parallel DE$,$\angle BAC = 35^o$ and $\angle CDE = 53^o$,find $\angle DCE$. (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 $\angle PQR = \angle PRQ$,then prove that $\angle PQS = \angle PRT$.

In the figure,lines $AB$ and $CD$ intersect at $O$. If $\angle AOC + \angle BOE = 70^o$ and $\angle BOD = 40^o$,find $\angle BOE$ and reflex $\angle COE$.

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

In the figure,sides $QP$ and $RQ$ of $\Delta PQR$ are produced to points $S$ and $T$ respectively. If $\angle SPR = 135^o$ and $\angle PQT = 110^o$,find $\angle PRQ$. (in $^o$)