Question

planes x and y intersect at a right angle. (overleftrightarrow{ab}) and (overleftrightarrow{cg}) lie in plane x and do not intersect. (overleftrightarrow{rs}) lies in plane y. which statements are true? select three options. (square) (overleftrightarrow{ab}) and (overleftrightarrow{cg}) are parallel. (square) (overleftrightarrow{ab}) and (overleftrightarrow{rs}) are parallel. (square) (overleftrightarrow{cg}) and (overleftrightarrow{rs}) are perpendicular. (square) (overleftrightarrow{ab}) and (overleftrightarrow{rs}) must intersect. (square) (overleftrightarrow{cg}) lies in plane x. (square) (overleftrightarrow{rs}) lies in plane x.

Explanation:

• For \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CG}\): Since \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CG}\) lie in plane \(X\) and do not intersect, by the definition of parallel lines (two lines in the same plane that do not intersect), \(\overleftrightarrow{AB}\parallel\overleftrightarrow{CG}\).
• For \(\overleftrightarrow{CG}\) and \(\overleftrightarrow{RS}\): Planes \(X\) and \(Y\) intersect at a right - angle. \(\overleftrightarrow{CG}\) lies in plane \(X\) and \(\overleftrightarrow{RS}\) lies in plane \(Y\). When two planes are perpendicular, a line in one plane and a line in the other plane (not parallel to the line of intersection of the planes) are perpendicular. Here, \(\overleftrightarrow{CG}\perp\overleftrightarrow{RS}\).
• For \(\overleftrightarrow{CG}\) lies in plane \(X\): Given in the problem statement “\(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CG}\) lie in plane \(X\)”.

• For \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{RS}\): \(\overleftrightarrow{AB}\) lies in plane \(X\) and \(\overleftrightarrow{RS}\) lies in plane \(Y\). Since the planes are perpendicular and the lines are not in a position to intersect (one is vertical - like in plane \(X\) and the other is horizontal - like in plane \(Y\)), they do not intersect.
• For \(\overleftrightarrow{RS}\) lies in plane \(X\): Given that \(\overleftrightarrow{RS}\) lies in plane \(Y\) (not plane \(X\)).

Answer:

\(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CG}\) are parallel; \(\overleftrightarrow{CG}\) and \(\overleftrightarrow{RS}\) are perpendicular; \(\overleftrightarrow{CG}\) lies in plane \(X\)