In the image, the question asks to determine which lines are parallel, using specific angle theorems as justification. The diagram consists of two intersecting lines forming angles with transversal lines.

Let's analyze the options and the diagram:

• The Corresponding Angles Theorem states that if two parallel lines are cut by a transversal, then the corresponding angles are congruent.
• The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
• The Same-Side Interior Angles Theorem (or Consecutive Interior Angles) states that if two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary.

Based on the diagram, the transversal $\overrightarrow{FK}$ crosses two lines, $\overrightarrow{FE}$ and $\overrightarrow{KM}$. To determine which lines are parallel, we need to examine the relationships between the angles formed by this transversal and the given lines.

• Option A claims $FK \parallel EM$ using the Converse of Corresponding Angles Theorem.
• Option B claims $FE \parallel FK$ using the Converse of the Alternate Interior Angles Theorem.
• Option C claims $FE \parallel KM$ using the Converse of the Same-Side Interior Angles Theorem.
• Option D claims $FE \parallel EM$ using the Converse of the Alternate Interior Angles Theorem.
• Option E claims $KM \parallel EK$ using the Converse of Same-Side Interior Angles Postulate.

Conclusion:

Looking at the diagram, the two lines that appear to be parallel are $FE$ and $KM$, as they seem to follow the conditions of the Alternate Interior Angles Theorem (Option D), implying that these two lines are parallel due to the angles formed between them and the transversal.

Let me know if you need further explanation or have additional questions!

Here are 5 related questions for deeper understanding:

1. How does the Converse of Corresponding Angles Theorem prove parallelism?
2. What is the difference between alternate interior angles and same-side interior angles?
3. Can two lines that are not parallel have congruent alternate interior angles?
4. How would the diagram change if the transversal was not perpendicular to one of the lines?
5. Why is it important for a transversal to cross two lines in proving parallelism?

Tip: When working with parallel lines and transversals, always start by identifying congruent or supplementary angles to apply the correct theorems.