$\angle ABC$ and $\angle ABD$ are a linear pair of angles. If $\angle ABC : \angle ABD = 17 : 13$,then find $\angle ABC$ and $\angle ABD$.

$A$ transversal intersects two lines in such a way that the two interior angles on the same side of the transversal are equal. Will the two lines always be parallel? Give reason for your answer.

In $\Delta ABC$,$\angle B = 70^{\circ}$ and $\angle C = 80^{\circ}$. The bisector of $\angle A$ intersects $BC$ at $D$. Find $\angle ADB$ and $\angle ADC$.

In the following figure,lines $AB$ and $CD$ intersect each other at point $P$. If $\angle APC : \angle BPC = 7 : 8$,find all the angles.

In the figure,if $AB \parallel CD \parallel EF$,$PQ \parallel RS$,$\angle RQD = 25^{\circ}$ and $\angle CQP = 60^{\circ}$,then $\angle QRS$ is equal to: (in $^{\circ}$)

In $\Delta ABC$,the sides $AB$ and $AC$ are produced to $D$ and $E$ respectively,so that exterior angles $\angle CBD$ and $\angle BCE$ are formed. If bisectors of $\angle CBD$ and $\angle BCE$ intersect at point $O$,prove that $\angle BOC = 90^{\circ} - \frac{1}{2} \angle A$.