The given diagram presents a set of intersecting lines, and the question asks which two statements are true regarding parallelism (denoted by "∥") or perpendicularity (denoted by "⊥") among certain line segments.

Step-by-step breakdown:

1. Identify the lines:

   • $\overleftrightarrow{FG}$, $\overleftrightarrow{HB}$, $\overleftrightarrow{AC}$, $\overleftrightarrow{DE}$ are labeled, and we need to determine their relationships based on the diagram.

2. Examine perpendicularity:

   • If lines are perpendicular (⊥), they form a 90° angle at the intersection.

3. Examine parallelism:

   • If lines are parallel (∥), they never intersect and maintain equal distance between them.

Analyzing each option:

• First option: $\overleftrightarrow{FG} \perp \overleftrightarrow{HB}$ and $\overleftrightarrow{AC} \perp \overleftrightarrow{HB}$

  • In the diagram, neither $\overleftrightarrow{FG}$ nor $\overleftrightarrow{AC}$ appears to form 90° angles with $\overleftrightarrow{HB}$.

• Second option: $\overleftrightarrow{FG} \parallel \overleftrightarrow{DE}$ and $\overleftrightarrow{DE} \parallel \overleftrightarrow{AC}$

  • From the diagram, $\overleftrightarrow{FG}$ and $\overleftrightarrow{DE}$ appear to be parallel as they maintain the same direction.
  • $\overleftrightarrow{DE} \parallel \overleftrightarrow{AC}$ is also likely true as the two lines are shown to be parallel.

• Third option: $\overleftrightarrow{HB} \perp \overleftrightarrow{AC}$ and $\overleftrightarrow{FG} \parallel \overleftrightarrow{DE}$

  • This does not seem valid because $\overleftrightarrow{HB}$ is not perpendicular to $\overleftrightarrow{AC}$.

• Fourth option: $\overleftrightarrow{FG} \perp \overleftrightarrow{HB}$ and $\overleftrightarrow{FG} \parallel \overleftrightarrow{DE}$

  • $\overleftrightarrow{FG}$ and $\overleftrightarrow{DE}$ are parallel but $\overleftrightarrow{FG}$ is not perpendicular to $\overleftrightarrow{HB}$.

Conclusion:

The second option, $\overleftrightarrow{FG} \parallel \overleftrightarrow{DE}$ and $\overleftrightarrow{DE} \parallel \overleftrightarrow{AC}$, appears to be the correct choice based on the diagram's geometry.

Would you like more details or explanations about parallelism and perpendicularity? Here are five related questions to deepen understanding:

1. How do we determine if two lines are parallel in a diagram?
2. What properties define perpendicular lines?
3. Can two non-intersecting lines always be assumed parallel?
4. How does the transitive property of parallelism apply here?
5. Can a pair of lines be both perpendicular and parallel at different points?

Tip: When working with geometric diagrams, pay close attention to the angles formed at intersections to determine perpendicularity.