To solve the question of determining which statement is true given that [tex]$\overrightarrow{PQ}$[/tex] and [tex]$\overrightarrow{RS}$[/tex] intersect to form four right angles, we need to understand the geometric properties involved.
1. Understanding Right Angles:
When two lines intersect to form right angles, it means that the angle between the lines is [tex]$90^\circ$[/tex]. Four right angles form around the point of intersection because the total angle around a point is [tex]$360^\circ$[/tex], and [tex]$360^\circ$[/tex] divided by [tex]$90^\circ$[/tex] gives 4 right angles.
2. Types of Intersecting Lines:
- Two lines can be either parallel, skew, or perpendicular.
- Parallel lines never intersect and maintain a constant distance between each other.
- Skew lines do not intersect and are not parallel; they exist in different planes.
- Perpendicular lines intersect at a [tex]$90^\circ$[/tex] angle.
Given that [tex]$\overrightarrow{PQ}$[/tex] and [tex]$\overrightarrow{RS}$[/tex] intersect and form four right angles, this intersection property indicates that they form [tex]$90^\circ$[/tex] angles with each other. Specifically, this means:
[tex]$\overrightarrow{PQ} \perp \overleftrightarrow{RS}$[/tex]: The two lines are perpendicular to each other.
3. Evaluating Statements:
- Option A: [tex]$\overrightarrow{PQ}$[/tex] and [tex]$\overrightarrow{RS}$[/tex] are parallel.
- This is not true because parallel lines never intersect.
- Option B: [tex]$\overrightarrow{PQ}=\overleftrightarrow{RS}$[/tex]
- This implies that both lines are the same, but intersecting lines are not the same line.
- Option C: [tex]$\overrightarrow{PQ}$[/tex] and [tex]$\overrightarrow{RS}$[/tex] are skew.
- This is not true because skew lines do not intersect, while we are given that the lines intersect.
- Option D: [tex]$\overrightarrow{PQ} \perp \overleftrightarrow{RS}$[/tex]
- This is true because they intersect to form right angles, which by definition means they are perpendicular.
Thus, the correct statement is:
Option D: [tex]$\overrightarrow{PQ} \perp \overleftrightarrow{RS}$[/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
