We are given a diagram with two lines. The first line is labeled as $P$ on the left and has an arrow in the middle suggesting its direction. The second line is labeled as $S$ on the right and also has an arrow in the middle suggesting its direction. These two lines are parallel. There is also a transversal that intersects these two parallel lines. The points of intersection on the transversal are named as $Q$, $R$, and $Z$. We need to find the relation between angles $x$ and $y$, where the angles are labelled as $\angle PQR = x$, $\angle QRS = y$, and $\angle Z = \angle QZR$.

1. Problem Description

We are given a diagram with two lines. The first line is labeled as $P$ on the left and has an arrow in the middle suggesting its direction. The second line is labeled as $S$ on the right and also has an arrow in the middle suggesting its direction. These two lines are parallel. There is also a transversal that intersects these two parallel lines. The points of intersection on the transversal are named as $Q$, $R$, and $Z$. We need to find the relation between angles $x$ and $y$, where the angles are labelled as $\angle PQR = x$, $\angle QRS = y$, and $\angle Z = \angle QZR$.

2. Solution Steps

Since the lines $PQ$ and $RS$ are parallel, we know some angle relationships hold true.

$\angle PQR$ and $\angle QRS$ are co-interior angles. The line segment $QR$ is a transversal of the two parallel lines $PQ$ and $RS$.

The sum of co-interior angles is $180$ degrees. Therefore, $\angle PQR + \angle QRS = 180^\circ$.

$x + \angle QRS = 180^{\circ}$.

Since $Z$ is the angle formed by the transversal and segment $QR$, therefore the sum of the angles of the triangle $QZR$ is $180^\circ$.

In triangle $QZR$, we have $\angle Z + x + y = 180^{\circ}$.

Since $\angle QZR$ is an exterior angle to the triangle formed by $Q, R, Z$ we have $\angle Z = x+y$. We can write:

$x+y+x+y = 180$.

$2x+2y=180$.

We have that $x$ and $y$ are interior angles of triangle QZR and hence must satisfy:

$x + y < 180$.

However, without more information about angle Z, we cannot simplify further.

3. Final Answer

The sum of angles $x$ and $y$ must be less than 180 degrees. $x + y < 180$.

Also, it follows that $2x + 2y = 180$ is incorrect.

Final Answer: $x + y + z = 180$ and $z + y = x$.

The relationship is $z = x+y$. Also $x + y + z = 180$.