1. The problem states that we have a triangle with three interior angles: $20^\circ$, $3x$, and $x$. We need to write an equation for the sum of these angles and then find the value of $x$. 2. The sum of the interior angles in any triangle is always $180^\circ$. This is a fundamental rule in geometry. 3. Using this rule, we write the equation: $$20 + 3x + x = 180$$ 4. Combine like terms: $$20 + 4x = 180$$ 5. Subtract 20 from both sides: $$\cancel{20} + 4x - \cancel{20} = 180 - 20$$ $$4x = 160$$ 6. Divide both sides by 4 to solve for $x$: $$\frac{4x}{\cancel{4}} = \frac{160}{\cancel{4}}$$ $$x = 40$$ 7. Therefore, the value of $x$ is $40$ degrees.