The problem in the image involves a geometric proof with the given conditions and a diagram showing two intersecting lines. The goal is to prove that two lines, labeled as $a$ and $b$, are parallel, i.e., $a \parallel b$.

Given Information:

1. $\angle 1 \cong \angle 7$ (these angles are congruent).
2. $c \parallel d$ (lines $c$ and $d$ are parallel).

Plan of Proof:

• To prove that $a \parallel b$, we will rely on properties of parallel lines and corresponding/congruent angles.

Proof Layout:

1. Statement: $\angle 1 \cong \angle 7$

   • Reason: Given.

2. Statement: $c \parallel d$

   • Reason: Given.

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

   • Reason: Corresponding angles are congruent (because $c \parallel d$ and $\angle 1$ and $\angle 5$ are corresponding angles).

4. Statement: $\angle 5 \cong \angle 7$

   • Reason: Transitive property of congruence (if $\angle 1 \cong \angle 5$ and $\angle 1 \cong \angle 7$, then $\angle 5 \cong \angle 7$).

5. Statement: $a \parallel b$

   • Reason: If alternate interior angles are congruent, then the lines are parallel. Since $\angle 5 \cong \angle 7$, this proves that $a \parallel b$.

This completes the proof that $a \parallel b$.

Would you like to explore this further or see more details about any specific steps? Here are five related questions to deepen your understanding:

1. Why are corresponding angles congruent when two lines are parallel?
2. What is the transitive property of congruence?
3. Can you always prove that two lines are parallel by using alternate interior angles?
4. How would the proof change if the given condition was $\angle 1 = \angle 6$ instead?
5. What are other ways to prove that lines are parallel in geometry?

Tip: Always check the diagram carefully to see which angles are corresponding, alternate interior, or vertical to apply the correct theorem.