The problem asks us to find the arc $y$ that corresponds to the central angle of $50^\circ$ given the arc $x$ is $50^\circ$.

1. Problem Description

The problem asks us to find the arc $y$ that corresponds to the central angle of $50^\circ$ given the arc $x$ is $50^\circ$.

2. Solution Steps

The degree measure of an arc is equal to the measure of its central angle. In this problem, the central angle is given as $50^\circ$.

The circumference of a circle is $360^\circ$. The two arcs, $x$ and $y$, combine to form the entire circle. Therefore, we have

$x + y = 360^\circ$

We are given that the central angle for arc $x$ is $50^\circ$. Since the degree measure of the arc $x$ is equal to the measure of its central angle, $x = 50^\circ$.

Therefore, we can substitute $x = 50^\circ$ in the equation $x + y = 360^\circ$:

$50^\circ + y = 360^\circ$

$y = 360^\circ - 50^\circ$

$y = 310^\circ$

3. Final Answer

$y = 310^\circ$