Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If two lines \(L_{1}\) and \(L_{2}\) are parallel to a plane \(P\), then \(L_{1}\) and \(L_{2}\) are parallel.

Short answer

The statement is False. Two lines parallel to same plane are not necessarily parallel to each other. They could be intersecting or skew lines as well.

Step-by-step solution

Understand the concept of parallelism

Two lines or planes are said to be parallel if they by definition never intersect. When applied to three-dimensional geometry, if two lines are parallel to a plane, they lie in the plane or are equidistant from the plane.

Apply the concept to the given problem

In this case, lines \(L_{1}\) and \(L_{2}\) are both parallel to plane \(P\). Hence, either they are on the plane or at a constant distance from it. However, this does not guarantee that \(L_{1}\) and \(L_{2}\) are parallel to each other. They can be like railway tracks (parallel) or they can intersect at some point in the space, outside the plane.

Decide if statement is true or false

Based on the above analysis, we can see that the given statement 'If two lines \(L_{1}\) and \(L_{2}\) are parallel to a plane \(P\), then \(L_{1}\) and \(L_{2}\) are parallel' is not necessarily true. The lines could possibly intersect or be skew, meaning they are non-parallel and non-intersecting but are not in the same plane.