The problem asks to find the value of angle $x$. We are given a triangle with two interior angles, $82^\circ$ and $67^\circ$. The angle $x$ is the exterior angle adjacent to the $67^\circ$ angle.

1. Problem Description

The problem asks to find the value of angle $x$. We are given a triangle with two interior angles, $82^\circ$ and $67^\circ$. The angle $x$ is the exterior angle adjacent to the $67^\circ$ angle.

2. Solution Steps

First, we find the third interior angle of the triangle. The sum of the angles in a triangle is $180^\circ$. Let the third interior angle be $y$. Then we have

$82^\circ + 67^\circ + y = 180^\circ$

$149^\circ + y = 180^\circ$

$y = 180^\circ - 149^\circ$

$y = 31^\circ$

Since $x$ is an exterior angle of the triangle, it is equal to the sum of the two non-adjacent interior angles.

$x = 82^\circ + 67^\circ$

$x = 149^\circ$

Alternatively, the exterior angle and the adjacent interior angle are supplementary. Therefore,

$x + 67^\circ = 180^\circ$

$x = 180^\circ - 67^\circ$

$x = 113^\circ$

The third interior angle is $y = 180^\circ - 67^\circ - 82^\circ = 31^\circ$.

The exterior angle $x$ is supplementary to the $31^\circ$ angle.

$x + 31^\circ = 180^\circ$

$x = 180^\circ - 31^\circ$

$x = 149^\circ$

3. Final Answer

149