In Fig. 6.13, lines AB and CD intersect at O. If $\angle AOC + \angle BOE = 70^{\circ}$ and $\angle BOD = 40^{\circ}$, find $\angle BOE$ and reflex $\angle COE$.

In Fig. 6.15, $\angle PQR = \angle PRQ$, then prove that $\angle PQS = \angle PRT

In Fig. 6.14, lines XY and MN intersect at O. If $\angle POY = 90^{\circ}$ and $a : b = 2 : 3$, find $c$.

It is given that $\angle XYZ = 64^{\circ}$ 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 $\angle QYP$.

In Fig. 6.17, 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 Fig. 6.26, if $AB \parallel CD$, $\angle APQ = 50^{\circ}$ and $\angle PRD = 127^{\circ}$, find $x$ and $y$.

In Fig. 6.27, 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 $AB \parallel CD$

In Fig. 6.23, if $AB \parallel CD$, $CD \parallel EF$ and $y : z = 3 : 7$, find $x$.

In Fig. 6.25, if $PQ \parallel ST$, $\angle PQR = 110^{\circ}$ and $\angle RST = 130^{\circ}$, find $\angle QRS$. [Hint : Draw a line parallel to ST through point R

In Fig. 6.24, if $AB \parallel CD$, $EF \perp CD$ and $\angle GED = 126^{\circ}$, find $\angle AGE$, $\angle GEF$ and $\angle FGE$.

In Fig. 6.16, if $x + y = w + z$, then prove that AOB is a line.