If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio $2:3,$ then the greater of the two angles is (in $^{\circ}$)

$A$ 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 $\Delta PQR$,$\angle P : \angle Q : \angle R = 5 : 2 : 2$. Find the measure of each angle of $\Delta PQR$.

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.

$\angle ACD$ is an exterior angle of $\Delta ABC$. The bisector of $\angle ABC$ and exterior $\angle ACD$ intersect each other at point $E$. Prove that,$\angle BEC = \frac{1}{2} \angle BAC$.

In the following figure,ray $YW$ stands on a line $XYZ$. $\angle WYZ : \angle WYX = 1 : 2$. Ray $YQ$ and ray $YP$ are angle bisectors of $\angle WYZ$ and $\angle WYX$ respectively. Find $\angle PYQ$. (in $^{\circ}$)