We are given a triangle with one interior angle of $31^{\circ}$. The exterior angle at one vertex is $284^{\circ}$, and the interior angle at that vertex is labeled $x$. We need to find the values of $x$ and $y$, where $y$ is the remaining interior angle.

1. Problem Description

We are given a triangle with one interior angle of $31^{\circ}$. The exterior angle at one vertex is $284^{\circ}$, and the interior angle at that vertex is labeled $x$. We need to find the values of $x$ and $y$, where $y$ is the remaining interior angle.

2. Solution Steps

First, we find the value of $x$. Since the sum of an interior angle and its exterior angle is $180^{\circ}$, we have:

$x + 284^{\circ} = 360^{\circ}$

$x = 360^{\circ} - 284^{\circ}$

$x = 76^{\circ}$

Next, we find the value of $y$. The sum of the interior angles of a triangle is $180^{\circ}$. Therefore,

$31^{\circ} + x + y = 180^{\circ}$

$31^{\circ} + 76^{\circ} + y = 180^{\circ}$

$107^{\circ} + y = 180^{\circ}$

$y = 180^{\circ} - 107^{\circ}$

$y = 73^{\circ}$

3. Final Answer

$x = 76^{\circ}$

$y = 73^{\circ}$