In the following figure, $AB$ is parallel to $CD$. $P$ and $R$ are the points on $AB$ and $CD$, respectively. $Q$ is in between $AB$ and $CD$. Find the value of $x$ in degrees.

In the figure below (not to scale), $ABC$ is a straight line. If $\angle FBE=60°, \angle CBG=120°, \angle ABG=x°$, $\angle ABF=\gamma °$ and $\angle CBE=z°=2\gamma °$, then $\left(x°+z°\right):y°$ is

In the figure below, $\overline{EF}\left|\right|\overline{GH}$. If $\overline{GI}$ is the transversal, $\angle IGH=y°$ and $\angle FIG=3y°$, then the ratio of the supplement of $y$ to its complement is

In the figure below (not to scale), $\overline{CD}\left|\right|\overline{RS}, \angle EMG=90°$, $\angle GMD=y°, \angle CME=x°$ and $y°=\frac{x°}{2}$. Then, $\angle FNS:\angle FNR$ is

In the above figure, $DF$ is parallel to $MN$. $EGH$ is an isosceles triangle, where $EG=EH$ and $\angle GEH=50°$. If $EM$ and $EN$ are the bisectors of the $\angle DEG$ and $\angle EFH$, then

Show that $\angle DEM=\angle FEN$.

In the above figure, $DF$ is parallel to $MN$. $EGH$ is an isosceles triangle, where $EG=EH$ and $\angle GEH=50°$. If $EM$ and $EN$ are the bisectors of the $\angle DEG$ and $\angle EFH$, then

Show that $\angle GEM=\angle HEN$.

In the above $△ABC$(not to scale), $OA$ is the angle bisector of $\angle BAC$. If $OB=OC$, then $\angle OAC=40°$ and $\angle ABO=20°$. If $\angle OCB=\frac{1}{2}\angle ACO$, then find $\angle BOC.$

In the given figure, $AB$ and $DE$ are straight lines. $\angle BAC=40°, \angle BPD=110°$ and $\angle DEC=40°$. Find $\angle ACE.$

In the following figure, $ABCD$ is a parallelogram. Find the value of $r.$