Question

use the photo to decide whether the statement is true or false. explain your reasoning.
a. the plane containing the floor of the tree house is parallel to the ground.
true; the floor is level with the vertical just like the ground.
false; the plane containing the floor is not on the same plane as the ground.
true; the floor is level with the horizontal just like the ground.
false; the plane containing the floor is above the plane containing the ground.
b. the lines containing the railings of the staircase, such as \\( \overleftrightarrow { a b } \\), are skew to all lines in the plane containing the ground
true: the lines intersect the plane of the ground, so they intersect certain lines of that plane.
true; the lines do not intersect the plane of the ground, so they do not intersect any lines of that plane.
false; the lines intersect the plane of the ground, so they intersect certain lines of that plane.
false; the lines do not intersect the plane of the ground, so they do not intersect any lines of that plane

Explanation:

Part a

• Understanding Parallel Planes:
• Parallel planes are planes that do not intersect and are always the same distance apart.
• The ground is a horizontal plane. The floor of the tree - house is also a horizontal plane.
• Horizontal planes are parallel to each other.
• The statement “The plane containing the floor of the tree house is parallel to the ground” is True.
• The first option “true; The floor is level with the vertical just like the ground” is incorrect because being level with the vertical has no relation to parallelism.
• The second option “false; The plane containing the floor is not on the same plane as the ground” is incorrect. Parallel planes are not the same plane.
• The fourth option “false; The plane containing the floor is above the plane containing the ground” is incorrect. Just because one plane is above another does not mean they are not parallel.
• The third option “true; The floor is level with the horizontal just like the ground” is correct as both the floor and the ground are horizontal planes (and horizontal planes are parallel)

Part b

• Understanding Skew Lines:
• Skew lines are non - intersecting and non - parallel lines that do not lie in the same plane.
• The lines containing the railings of the staircase (like \(\overleftrightarrow{AB}\)) intersect the plane of the ground.
• If a line intersects a plane, it intersects at least one line in that plane.
• Skew lines do not intersect the plane.
• The first option “true: The lines intersect the plane of the ground, so they intersect certain lines of that plane” is incorrect because if a line intersects a plane, it cannot be skew to all lines in that plane.
• The second option “true; The lines do not intersect the plane of the ground, so they do not intersect any lines of that plane” is incorrect because the lines (railings) do intersect the ground plane.
• The fourth option “false; The lines do not intersect the plane of the ground, so they do not intersect any lines of that plane” is incorrect as the lines intersect the ground plane.
• The third option “false; The lines intersect the plane of the ground, so they intersect certain lines of that plane” is correct. Since the lines intersect the plane of the ground, they are not skew to all lines in the ground plane

Answer:

a. true; The floor is level with the horizontal just like the ground
b. false; The lines intersect the plane of the ground, so they intersect certain lines of that plane