To solve the problem, let's break it down step by step using the given information:

Given:

1. $\overline{BC} \perp \overline{CD}$: Line $BC$ is perpendicular to $CD$.
2. $\angle 1$ and $\angle 2$ are complementary.

To Prove:

$\overline{BA} \parallel \overline{CD}$.

Proof:

1. Perpendicularity Property: Since $\overline{BC} \perp \overline{CD}$, $\angle BCD = 90^\circ$.

2. Definition of Complementary Angles: By the given, $\angle 1 + \angle 2 = 90^\circ$.

3. Transversals and Parallel Lines: In the diagram:

   • $\overline{AC}$ acts as a transversal.
   • $\angle 1$ is an alternate interior angle relative to $\angle BCD$ (due to the line $\overline{AC}$).
   • $\angle 2$ lies on the opposite side of the transversal and is complementary to $\angle BCD$.

4. Converse of the Corresponding Angles Theorem: If $\angle 1 + \angle 2 = 90^\circ$, and one of these angles corresponds to the perpendicular relationship between $\overline{BC}$ and $\overline{CD}$, then $\overline{BA} \parallel \overline{CD}$.

Thus, $\overline{BA} \parallel \overline{CD}$ is proven.

Would you like further clarification on any of these steps?

Related Questions:

1. What is the importance of complementary angles in determining parallel lines?
2. How does the perpendicularity of lines relate to complementary angles?
3. Can this proof be extended to other geometric shapes like triangles?
4. How would the proof change if the lines weren’t perpendicular?
5. Why is the transversal $\overline{AC}$ critical in this proof?

Tip:

Always look for relationships between angles and lines (e.g., complementary, supplementary, corresponding, alternate interior) to prove parallel or perpendicular lines.