The problem involves two parallel lines and a transversal that forms two triangles. We need to determine the relationship between angles $x$, $y$, and $z$.

1. Problem Description

The problem involves two parallel lines and a transversal that forms two triangles. We need to determine the relationship between angles $x$, $y$, and $z$.

2. Solution Steps

Since lines $PQ$ and $RS$ are parallel, the angle at $Q$ (labeled $x$) and the angle at $R$ (labeled $y$) are alternate interior angles. These angles are therefore equal.

$x = y$

Now consider triangle $QZR$. The sum of the interior angles in a triangle is $180^\circ$. Therefore, in triangle $QZR$, we have:

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

Since $x = y$, we can substitute $x$ for $y$ in the equation above:

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

$2x + z = 180^\circ$

Therefore $2y + z = 180^\circ$

3. Final Answer

The relationship is $2x + z = 180^\circ$ and $x=y$. This is equivalent to $2y + z = 180^\circ$.