1. **State the problem:** We need to find the value of the angle $x$ in a circle where two secants intersect outside the circle, creating angles labeled $20^\circ$, $104^\circ$, and $x^\circ$. 2. **Formula used:** When two secants intersect outside a circle, the measure of the angle formed is half the difference of the measures of the intercepted arcs. Mathematically, $$x = \frac{|\text{arc}_1 - \text{arc}_2|}{2}$$ 3. **Identify arcs:** Given arcs are $20^\circ$ and $104^\circ$. 4. **Calculate difference of arcs:** $$|104 - 20| = 84$$ 5. **Calculate angle $x$:** $$x = \frac{84}{2} = 42$$ 6. **Conclusion:** The value of $x$ is $42^\circ$. This matches the given value, confirming the calculation.