1. **State the problem:** We are given a trapezoid with non-parallel sides equal, and two triangles with given side lengths and angles. We need to find the value of $x$. 2. **Analyze the trapezoid:** The trapezoid has non-parallel sides equal, so it is an isosceles trapezoid. The non-parallel sides are $3x - 2$ and $2x + 17$. Since they are equal, set: $$3x - 2 = 2x + 17$$ 3. **Solve for $x$:** $$3x - 2 = 2x + 17$$ $$3x - \cancel{2} = 2x + 17$$ Subtract $2x$ from both sides: $$3x - \cancel{2x} - 2 = \cancel{2x} + 17$$ $$x - 2 = 17$$ Add $2$ to both sides: $$x - \cancel{2} + 2 = 17 + 2$$ $$x = 19$$ 4. **Check the triangles:** The triangles have sides $20$, $24$ and $x + 2$, $x$ respectively. With $x=19$, the sides are: $$x + 2 = 19 + 2 = 21$$ $$x = 19$$ 5. **Summary:** The value of $x$ that satisfies the trapezoid side equality is: $$\boxed{19}$$