Two lines parallel to a plane are parallel: True False Two lines perpendicular to plane are parallel: True False (g) Two planes parallel to line are parallel. True False (h) Two planes perpendicular to line are parallel: True False Two planes either intersect or are parallel: True False Two lines either intersect or are parallel: True False k) A plane and line either intersect or are parallel: True False
The Correct Answer and Explanation is:
Let’s break down each statement and determine whether it’s true or false:
- Two lines parallel to a plane are parallel:
False. Two lines that are parallel to a plane can be either parallel to each other or not. The fact that two lines are parallel to the same plane does not guarantee that the lines themselves are parallel to each other. The lines may be skewed (i.e., not intersecting but not parallel either). - Two lines perpendicular to a plane are parallel:
True. If two lines are perpendicular to the same plane, they must be parallel. This is because they both lie along the same direction relative to the plane and have the same orientation. - Two planes parallel to a line are parallel:
True. If two planes are parallel to the same line, they are parallel to each other. The parallelism to the same line ensures that the planes do not intersect, and hence must be parallel. - Two planes perpendicular to a line are parallel:
False. Two planes that are both perpendicular to the same line are not necessarily parallel. The planes can be oriented differently in 3D space, potentially intersecting each other along a line that is perpendicular to the original line. - Two planes either intersect or are parallel:
True. In Euclidean geometry, two planes either intersect along a line or are parallel. There is no other possibility for the relationship between two planes in a 3D space. - Two lines either intersect or are parallel:
True. In a 3D space, two lines must either intersect at one point or be parallel (i.e., they do not meet but are in the same direction). They cannot “bend” around each other, so this is true. - A plane and line either intersect or are parallel:
True. A line and a plane in 3D space will either intersect at one point or the line will be parallel to the plane. They can’t have any other relationship in Euclidean geometry.
Conclusion:
Most of these statements are based on fundamental geometric principles. Some of the answers depend on specific conditions, such as the orientation of lines and planes relative to each other. In 3D geometry, lines and planes have restricted relationships that can only be intersection, parallelism, or skewness, which makes some statements true and others false.
