Question

given ( a perp c ) and ( b perp c ), what can you say about ( a ) and ( b )? explain.
a. ( a perp b ); if two lines are perpendicular to the same line, then they are perpendicular to each other.
b. ( a ) and ( b ) form a straight line; if two lines are perpendicular to the same line, then they form a straight line.
c. ( a ) and ( b ) are intersecting lines; if two lines are perpendicular to the same line, then they are intersecting lines.
d. ( a parallel b ); if two lines are perpendicular to the same line, then they are parallel to each other.

Explanation:

To solve this, we recall the geometric principle: In a plane, if two lines are perpendicular to the same line, they are parallel to each other. Let's analyze each option:

• Option A: The statement "if two lines are perpendicular to the same line, then they are perpendicular to each other" is incorrect.
• Option B: Two lines perpendicular to the same line do not necessarily form a straight line. This is incorrect.
• Option C: Just being perpendicular to the same line doesn't mean they are only intersecting (they are parallel, a special case of non - intersecting in a plane, but in the context of this principle, they are parallel). So this is incorrect.
• Option D: Follows the correct geometric principle that if two lines (\(a\) and \(b\)) are perpendicular to the same line (\(c\)), then \(a\parallel b\).

Answer:

D. \(a \parallel b\); if two lines are perpendicular to the same line, then they are parallel to each other.