1. **State the problem:** We are given that \(\angle 1 \cong \angle 2\) and we want to prove that lines \(a\) and \(b\) are parallel. 2. **Given:** \(\angle 1 \cong \angle 2\). 3. By the **Vertical Angles Theorem**, \(\angle 1 \cong \angle 3\) because vertical angles are congruent. 4. Since \(\angle 1 \cong \angle 3\) and \(\angle 1 \cong \angle 2\), by transitive property, \(\angle 3 \cong \angle 2\). 5. \(\angle 3\) and \(\angle 2\) are alternate interior angles formed by the transversal crossing lines \(a\) and \(b\). 6. By the **Converse of the Alternate Interior Angles Theorem**, if alternate interior angles are congruent, then the lines \(a\) and \(b\) are parallel. **Therefore,** \(a \parallel b\) by the Converse of the Alternate Interior Angles Theorem. **Summary for the missing step in the proof:** 2. \(\angle 1 \cong \angle 3\) 3. \(a \parallel b\) by the Converse of the Alternate Interior Angles Theorem.