To determine which statement is true given that [tex]\(\overline{P Q}\)[/tex] and [tex]\(\overline{R S}\)[/tex] intersect to form four right angles, let's carefully analyze the properties of these geometric relationships:
1. Intersection and Right Angles:
- When two lines intersect to form four right angles, it means they intersect at a single point creating 90-degree angles between each pair of adjoining segments.
2. Understanding the Options:
- Option A: [tex]\(\overrightarrow{P Q} = \overrightarrow{R S}\)[/tex]
- This statement asserts that the two lines [tex]\(\overline{P Q}\)[/tex] and [tex]\(\overline{R S}\)[/tex] are equal in their vector representation. This would imply that both the lines overlap, but forming right angles at the point of intersection suggests the lines are distinct and meet at one point rather than being the same line.
- Option B: [tex]\(\overrightarrow{P Q}\)[/tex] and [tex]\(\overline{R S}\)[/tex] are parallel
- If two lines are parallel, they never intersect. Here, it is stated that they intersect and form right angles. Hence, they cannot be parallel.
- Option C: [tex]\(\overline{P Q}\)[/tex] and [tex]\(\overline{R S}\)[/tex] are skew
- Skew lines are lines that do not intersect and are not parallel, usually existing in different planes. Since it is given that these lines intersect, they cannot be skew.
- Option D: [tex]\(\overline{P Q} \perp \overrightarrow{R S}\)[/tex]
- Perpendicular lines intersect to form right angles. Since it is given that the lines [tex]\(\overline{P Q}\)[/tex] and [tex]\(\overline{R S}\)[/tex] intersect to form four right angles, they must be perpendicular to each other.
3. Conclusion:
- Given the definitions and understanding of the geometric relationships, we can conclude that the correct statement is:
[tex]\[ \overline{P Q} \perp \overrightarrow{R S} \][/tex]
Therefore, the correct answer is:
D. [tex]\(\overline{P Q} \perp \overrightarrow{R S}\)[/tex]
