The given diagram shows two intersecting lines, forming angles labeled 1, 2, 3, and 4. You are asked to prove that $\angle 1 \cong \angle 4$, given that $\angle 2 \cong \angle 3$.

Here's the step-by-step reasoning to prove this:

1. Vertical Angles Theorem: The angles formed by two intersecting lines are known as vertical angles, and vertical angles are congruent. From the diagram:

   • $\angle 1 \cong \angle 3$ (since they are vertical angles),
   • $\angle 2 \cong \angle 4$ (since they are also vertical angles).

2. Given Condition: We are provided with the information that $\angle 2 \cong \angle 3$.

3. Transitive Property of Congruence: Using the Transitive Property (if two angles are both congruent to a third angle, then they are congruent to each other), we can conclude:

   • Since $\angle 1 \cong \angle 3$ and $\angle 3 \cong \angle 2$, it follows that $\angle 1 \cong \angle 2$.

4. Vertical Angles Again: Finally, since $\angle 2 \cong \angle 4$ (by the Vertical Angles Theorem), and we’ve just shown $\angle 1 \cong \angle 2$, it follows that: $\angle 1 \cong \angle 4.$

Thus, we have proven that $\angle 1 \cong \angle 4$.

Do you need more details on any specific step or have any other questions?

Here are five questions that further explore this topic:

1. What is the Vertical Angles Theorem, and why does it hold for all intersecting lines?
2. How does the Transitive Property of Congruence apply to angles in general?
3. Can we use the same logic to prove that $\angle 2 \cong \angle 3$ is necessary for $\angle 1 \cong \angle 4$?
4. What other properties of intersecting lines can help in proving congruence of angles?
5. How would the solution change if we were given a different pair of congruent angles, such as $\angle 1 \cong \angle 3$?

Tip: Always remember that vertical angles formed by intersecting lines are congruent, which can be very useful in solving angle problems involving line intersections.