Q. In the figure, $\angle PQR=\angle PRQ$, then prove that $\angle PQS=\angle PRT$.

Q. In the figure, lines $XY$ and $MN$ intersect at $O$. If $\angle POY={90}^o$ and $a:b=2:3$, find $c$.

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

Q. 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
$\angle ROS=\frac{1}{2}(\angle QOS-\angle POS)$.

Q. It is given that $\angle XYZ={64}^{}$ and $XY$ is produced to point $P$. Draw a figure from the given information. If ray $YQ$ bisects $\angle ZYP$, find $\angle XYQ$ and reflex $\angle QYP$.

Q. In the figure, if $x+y=w+z$, then prove that $AOB$ is a line.

Q. In the figure, find the values of $x$ and $y$ and then show that $AB\parallel CD$.

Q. In the figure, if $AB\parallel CD,CD\parallel EF$ and $y:z=3:7$, find $x$.

Q. In the figure, $PQ$ and $RS$ are two mirrors placed parallel to each other. An incident ray $AB$ strikes the mirror $PQ$ at $B$, the reflected ray moves along the path $BC$ and strikes the mirror $RS$ at $C$ and again reflects back along $CD$. Prove that $AB\parallel CD$.

Q. In the figure, if $PQ\parallel ST,\angle PQR={110}^{}$ and $\angle RST={130}^{}$, find $\angle QRS$.

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

Q. In given fig. if $AB||CD,EF\perp CD$ and $\angle GED={126}^o$, find $\angle AGE,\angle GEF$ and $\angle FGE$.

Q. In the figure, if $AB\parallel DE,\angle BAC={35}^o$ and $\angle CDE={53}^o$, find $\angle DCE$.

Q. In Fig. 6.43, if $PQ\perp PS,PQ\parallel SR,\angle SQR={28}^0$ and $\angle QRT={65}^0$, then find the values of $x$ and $y$.

Q. 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 $\angle SQT$.

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

Q. 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 $\angle PRQ$.

Q. In the figure, $\angle X={62}^o,\angle XYZ={54}^o$. If $YO$ and $ZO$ are the bisectors of $\angle XYZ$ and $\angle XZY$ respectively of $\Delta XYZ$, find $\angle OZY$ and $\angle YOZ$.