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].
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].
