In the figures below, what angles do x and y stand for?

The angle $x$ and the $65^\circ$ angle are alternate interior angles. Since the two horizontal lines are parallel, alternate interior angles are equal.

$$x = 65^\circ$$

The angle corresponding to the $65^\circ$ angle is the interior angle on the top left. Let's call this angle $A$. So, $A = 65^\circ$.
Angle $y$ and angle $A$ form a linear pair (angles on a straight line), so their sum is $180^\circ$.

$$y + A = 180^\circ$$ $$y + 65^\circ = 180^\circ$$ $$y = 180^\circ - 65^\circ$$ $$y = 115^\circ$$

For Fig. 5.32 (Right Diagram):

The angle $x$ and the $53^\circ$ angle form a linear pair. The sum of angles in a linear pair is $180^\circ$.

$$x + 53^\circ = 180^\circ$$ $$x = 180^\circ - 53^\circ$$ $$x = 127^\circ$$

The angle $y$ and the $78^\circ$ angle are alternate interior angles. Since the two horizontal lines are parallel, alternate interior angles are equal.

$$y = 78^\circ$$

The angles are:
For the left diagram: $\boxed{x = 65^\circ, y = 115^\circ}$
For the right diagram: $\boxed{x = 127^\circ, y = 78^\circ}$