This problem asks to prove that $\angle 4 \cong \angle 3$ using the information provided and without relying on the transitive property for parallel lines. Let's break this down step by step.

Information Given:

1. $EF \parallel CD$
2. $\angle 8 \cong \angle 1$ (these angles are congruent)
3. $AB \parallel CD$

Steps for the Proof:

1. Statement: $EF \parallel CD$

   • Reason: Given.

2. Statement: $AB \parallel CD$

   • Reason: Given.

3. Statement: $\angle 8 \cong \angle 1$

   • Reason: Given.

4. Statement: $\angle 8 \cong \angle 4$

   • Reason: Corresponding angles are congruent because $EF \parallel CD$.

5. Statement: $\angle 1 \cong \angle 3$

   • Reason: Corresponding angles are congruent because $AB \parallel CD$.

6. Statement: $\angle 4 \cong \angle 3$

   • Reason: If $\angle 8 \cong \angle 4$ and $\angle 1 \cong \angle 3$, then by the transitive property (allowed here for $\cong$), $\angle 4 \cong \angle 3$.

This proof is primarily based on the fact that corresponding angles between parallel lines are congruent.

Do you have any further questions or would you like a detailed explanation for any of the steps?

Related Questions:

1. What is the difference between corresponding angles and alternate interior angles?
2. How can the transitive property be applied to angle congruency in parallel line proofs?
3. Can you explain how angle $4$ and angle $8$ are corresponding angles?
4. Why is it important that $AB \parallel CD$ and $EF \parallel CD$ in this proof?
5. Can vertical angles or other angle relationships be used in this proof?

Tip:

In geometry proofs, always clearly state what is "given" and how you apply properties like congruence or parallel lines. It helps to keep track of all relationships step by step.