1. **State the problem:** We have two chords RU and ST intersecting inside a circle, forming angles 13x and 3x + 40° respectively. We need to find the value of $x$. 2. **Formula used:** When two chords intersect inside a circle, the measure of each angle formed is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. However, since the problem gives the angles directly as expressions and they are vertical angles, they are equal. 3. **Set up the equation:** Since the angles are vertical angles, they are equal: $$13x = 3x + 40$$ 4. **Solve for $x$:** Subtract $3x$ from both sides: $$13x - \cancel{3x} = \cancel{3x} + 40 - 3x$$ $$10x = 40$$ Divide both sides by 10: $$\frac{10x}{\cancel{10}} = \frac{40}{\cancel{10}}$$ $$x = 4$$ 5. **Final answer:** $$\boxed{4}$$