Answered

If [tex]$\overrightarrow{P Q}$[/tex] and [tex]$\overrightarrow{R S}$[/tex] intersect to form four right angles, which statement is true?

A. [tex]$\overline{P Q}$[/tex] and [tex]$\overline{R S}$[/tex] are parallel
B. [tex]$\overline{P Q} = \overline{R S}$[/tex]
C. [tex]$\overline{P Q}$[/tex] and [tex]$\overrightarrow{R S}$[/tex] are skew
D. [tex]$\overline{P Q} \perp \overline{R S}$[/tex]



Answer :

To determine which statement is true if \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) intersect to form four right angles, we need to analyze the geometric relationships between the lines when they intersect perpendicularly.

1. Understanding Right Angles and Perpendicularity:
- When two lines intersect to form four right angles (90 degrees each), they are perpendicular to each other. This is a defining characteristic of perpendicular lines, as each of the intersecting angles will be exactly 90 degrees.

2. Analyzing the Statements:
- Option A: \(\overline{P Q}\) and \(\overline{R S}\) are parallel.
- Parallel lines never intersect. Since it is given that \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) intersect, this option cannot be true.

- Option B: \(\overline{P Q} = \overline{R S}\)
- This statement implies that the lengths of segments \(\overline{P Q}\) and \(\overline{R S}\) are equal. The lengths of the lines are not necessarily related to the angles they form upon intersection, so this option does not need to be true.

- Option C: \(\overline{P Q}\) and \(\overrightarrow{R S}\) are skew.
- Skew lines are lines that do not intersect and are not parallel (they exist in different planes). Given \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) intersect to form right angles, they cannot be skew.

- Option D: \(\overline{P Q} \perp \overline{R S}\)
- Perpendicular lines are defined by the fact that they intersect to form right angles. Given the condition that \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) form four right angles upon intersection, this implies that the lines are perpendicular.

3. Conclusion:
- The correct statement is: \(\overline{P Q} \perp \overline{R S}\).

Therefore, the true statement, given \(\overrightarrow{P Q}\) and \(\overrightarrow{R S}\) intersect to form four right angles, is:

D. [tex]\(\overline{P Q} \perp \overline{R S}\)[/tex].