1. The problem asks to find the value of $x$ in the triangle where the angles are $86^\circ$, $2x$, and $4x$. 2. Recall the Triangle Angle Sum Theorem: The sum of the interior angles of a triangle is always $180^\circ$. 3. Set up the equation using the given angles: $$86 + 2x + 4x = 180$$ 4. Combine like terms: $$86 + 6x = 180$$ 5. Subtract 86 from both sides: $$\cancel{86} + 6x - \cancel{86} = 180 - 86$$ $$6x = 94$$ 6. Divide both sides by 6 to solve for $x$: $$\frac{6x}{\cancel{6}} = \frac{94}{6}$$ $$x = \frac{94}{6} = 15.67$$ 7. Therefore, the value of $x$ is approximately $15.67^\circ$. This means the angles $2x$ and $4x$ are approximately $31.33^\circ$ and $62.67^\circ$ respectively, which together with $86^\circ$ sum to $180^\circ$ as expected.