We are given a diagram with two lines and angles labeled $x$, $y$, and $z$. The line segment $PQ$ is parallel to the line segment $RS$. We need to find the relation between angles $x, y,$ and $z$.

1. Problem Description

We are given a diagram with two lines and angles labeled $x$, $y$, and $z$. The line segment $PQ$ is parallel to the line segment $RS$. We need to find the relation between angles $x, y,$ and $z$.

2. Solution Steps

Let the line segment that connects points $Q$ and $R$ be $QR$.

Since $PQ$ is parallel to $RS$, the alternate interior angles formed by the transversal $QR$ are equal.

Let the angle between $QR$ and the line segment $PQ$ be $x$. The alternate interior angle, i.e., the angle between $QR$ and $RS$, must be equal to $x$. Thus, the angle between $QR$ and $RS$ is also $x$.

In the triangle $QRZ$, the sum of the angles is $180$ degrees. The angles are $x,y,$ and $z$ where the angle $x$ refers to the angle between $QR$ and $RS$.

Consider the angle $x$. This angle can be written as the sum of two smaller angles. The angles are $y$ and the interior angle of the triangle $QRZ$ at vertex $R$. Thus, the angle between line segment $QR$ and $RS$ is $x$. This implies that $y+z = x$.

Therefore, $x = y + z$.

3. Final Answer

$x = y+z$