Start with grouping the statements in all possible combinations and try to find an example for each group.
Statements I and III.
Practice makes perfect
We have been given the following three statements.
I. & l || m
II. & l and m do not intersect.
III. & l and m are skew.
In order to identify the two statements that contradict each other, we will first group the statements in all possible combinations.
Group A
Group B
Group C
I. l || m
I. l || m
II. l and m do not intersect.
II. l and m do not intersect.
III. l and m are skew.
III. l and m are skew.
Next, we will find an example for each group. If we cannot, this means that the two statements contradict each other. Let's start with Group A.
Group A
An example for Group A must be a pair of lines which are both parallel and do not intersect at the same time. Let's look at the following diagram.
As we can see, it is possible to find two lines l and m which are parallel and do not intersect at the same time. This means that statements I and II do not contradict each other.
Two lines are skew if they are noncoplanar; they are not parallel and they do not intersect.
As we can see, the very definition of two lines being skew tells us that they cannot be parallel. Therefore it is not possible to give an example of two lines which meet this requirements. Statements I and III contradict each other.
Group C
For the last group, we will start by drawing a rectangular prism.
Now, let's choose lines UQ and VW and name them l and m respectively.
As we can see, the lines l and m are clearly noncoplanar. Therefore, they are skew. Moreover, they do not intersect. Since we are able to give an example for Group C, statements II and III do not contradict each other.
