Determine the relationship between angles $x$ and $y$ in the given diagram, where the lines $PQ$ and $RS$ are parallel. The angles are labeled $x$ and $y$ respectively. There is also an angle labeled $z$.

1. Problem Description

Determine the relationship between angles $x$ and $y$ in the given diagram, where the lines $PQ$ and $RS$ are parallel. The angles are labeled $x$ and $y$ respectively. There is also an angle labeled $z$.

2. Solution Steps

The problem states that the lines $PQ$ and $RS$ are parallel.

We can observe that the angle $x$ and the angle adjacent to $z$ on the line joining $Q$ and $R$ are alternate interior angles. Because $PQ$ is parallel to $RS$, these two angles are congruent.

Let the angle adjacent to $z$ on line segment $QR$ be $a$. Then $a = x$.

Also, $a + z + y = 180^\circ$, because the angles $a$, $z$, and $y$ together form a straight line.

Since $a = x$, we can substitute $x$ for $a$ in the equation $a + z + y = 180^\circ$, to get $x + z + y = 180^\circ$.

The angles $x$, $y$ and $z$ are related by

$x + y + z = 180^{\circ}$.

3. Final Answer

$x + y + z = 180^\circ$