How many triangles can be drawn having its angles as $53^{\circ}, 64^{\circ}$ and $63^{\circ}$? Give reason for your answer.

In the following figure,ray $BA$ is perpendicular to line $CD$. If $x: y: z = 4: 5: 6$,then find the values of $x, y,$ and $z$.

$A$ 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.

$\angle X$ and $\angle Y$ are supplementary angles. If $\angle X : \angle Y = 23 : 13$,then find $\angle X$ and $\angle Y$.

In $\Delta ABC$,$\angle A + \angle B = 80^{\circ}$ and $\angle B + \angle C = 150^{\circ}$. Find the measure of each angle of $\Delta ABC$.

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