The question asks which statement could always be used to prove that line $\overline{AB} \parallel \overline{CD}$, given that transversal $EF$ intersects both lines $AB$ and $CD$ as shown in the figure.

Analyzing the diagram:

• The figure shows two lines $AB$ and $CD$, cut by transversal $EF$.
• The angles marked correspond to various relationships created by this transversal.

Answer Choices:

A. $\angle 2 \cong \angle 6$
B. $\angle 7 \cong \angle 8$
C. $\angle 3$ and $\angle 6$ are supplementary
D. $\angle 1$ and $\angle 5$ are supplementary

Understanding the geometric principles:

To prove two lines are parallel when a transversal cuts them, we use certain angle relationships. The most common are:

• Corresponding Angles Postulate: If two corresponding angles are congruent, then the lines are parallel.
• Alternate Interior Angles Theorem: If two alternate interior angles are congruent, then the lines are parallel.
• Same-Side Interior Angles Postulate: If the sum of two same-side interior angles is 180° (supplementary), then the lines are parallel.

Breakdown of the options:

• Option A: $\angle 2 \cong \angle 6$.
  These are corresponding angles. According to the Corresponding Angles Postulate, if these angles are congruent, then $\overline{AB} \parallel \overline{CD}$. This is the correct choice.

• Option B: $\angle 7 \cong \angle 8$.
  These appear to be two exterior angles, but they are not corresponding, alternate interior, or same-side angles in this context. This is incorrect.

• Option C: $\angle 3$ and $\angle 6$ are supplementary.
  These angles are on the same side of the transversal, but they are not interior angles between the two lines. This is incorrect.

• Option D: $\angle 1$ and $\angle 5$ are supplementary.
  While these angles are on the same side of the transversal, they are also not interior angles between the two lines, so this option is incorrect.

Correct Answer:

A. $\angle 2 \cong \angle 6$

This option uses corresponding angles, which can always be used to prove that two lines are parallel when cut by a transversal.

Do you want more details on this? Feel free to ask!

Here are 5 related questions for further exploration:

1. How can the Alternate Interior Angles Theorem be used to prove lines parallel?
2. What is the definition of corresponding angles in geometry?
3. How do you prove lines are parallel using supplementary angles?
4. What are same-side interior angles, and how are they related to parallel lines?
5. Why can't angles 7 and 8 prove that lines $AB$ and $CD$ are parallel?

Tip:

Always look for angle relationships (corresponding, alternate interior, same-side) when trying to prove two lines are parallel using a transversal!