The problem provides a diagram with lines and angles marked as $x$, $y$, and $z$. The line $PQ$ is parallel to $RS$. The problem asks to determine the relationship between angles $x, y, z$.

1. Problem Description

The problem provides a diagram with lines and angles marked as $x$, $y$, and $z$. The line $PQ$ is parallel to $RS$. The problem asks to determine the relationship between angles $x, y, z$.

2. Solution Steps

Since $PQ$ is parallel to $RS$, we can use the properties of parallel lines and transversals to relate the angles.

First, let's consider the line segment $QR$ as a transversal cutting the parallel lines $PQ$ and $RS$. Let $a$ be the angle between the line segment $QR$ and line $PQ$ and let $b$ be the angle between the line segment $QR$ and line $RS$.

Since $PQ$ and $RS$ are parallel, we know that the alternate interior angles are equal, so $a = b$.

Now, we can express $a$ and $b$ in terms of the given angles. From the triangle formed by points $Q, R, Z$, the sum of angles in a triangle is 180 degrees. Therefore, $x + y + z = 360^{\circ}$.

Now, consider extending segment $PQ$ past Q. The angle formed between $PQ$ and $QR$ is $a$, which means that the exterior angle to x along the line $PQ$ is $a$.

Since the exterior angle is equal to $180 - x$, we have $a = 180 - x$.

Similarly, consider extending segment $RS$ past $R$. The angle formed between $RS$ and $QR$ is $b$, which means the exterior angle is $y$. Since $RS$ and $QR$ add up to $b$, $b = 180 - y$.

However, since $PQ || RS$, it means the angles on the same side of the transversal add up to

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$a = 180 - x$ and $b = 180 - y$.

Since $a=b$, $180 - x = 180 - y$ so $x=y$.

Since $x+y+z$ forms a $360$ degree angle we cannot determine what the relation between $x,y,z$ are.

But since the transversal cut $PQ$ and $RS$, $x+z+y$ create a full revolution.

Also notice that angle $z$ is the third angle of the triangle with angles $x$ and $y$. So we can use that the sum of angles of the triangle equals $180$ degrees. $x + y +$ the angle at $z$ equal $180$ degrees.

Since the alternate angles $a$ and $b$ are equal where $a = b$ we can determine that $x = y$. Then $x+y+z = 360$.

The angle sum is $180$, So we can describe $z$ as $180 - x - y$.

Another possible approach is to consider that $z$ is the exterior angle to angles $x$ and $y$.

3. Final Answer

$x=y$