1. **Problem Statement:** Determine which statements can be used to prove that lines $a \parallel b$ and $c \parallel d$ based on angle relationships. 2. **Key Concept:** Lines are parallel if corresponding angles are congruent or if alternate interior angles are congruent, or if consecutive interior angles are supplementary. 3. **Given Options:** - $\angle 1 \cong \angle 6$, $\angle 3 \cong \angle 5$ - $\angle 1 \cong \angle 6$, $\angle 4$ and $\angle 5$ are supplementary - $\angle 1 \cong \angle 4$, $\angle 1$ and $\angle 2$ are supplementary - $\angle 1$ and $\angle 3$ are supplementary, $\angle 1$ and $\angle 6$ are supplementary 4. **Analysis:** - $\angle 1 \cong \angle 4$ indicates alternate interior angles are congruent, which is a direct criterion for parallel lines. - $\angle 1$ and $\angle 2$ being supplementary suggests they form a linear pair, confirming the straight line. 5. **Conclusion:** The statement "$\angle 1 \cong \angle 4$, $\angle 1$ and $\angle 2$ are supplementary" can be used to prove $a \parallel b$ and $c \parallel d$. **Final answer:** $\boxed{\angle 1 \cong \angle 4, \ \angle 1 \text{ and } \angle 2 \text{ are supplementary}}$