 Determine whether each statement is true or false in double-struck R3.
(a)
Two lines parallel to a third line are parallel.
True False 

The statement in question is about the properties of parallel lines in \(\mathbb{R}^3\) (the three-dimensional real coordinate space).

### (a) Two lines parallel to a third line are parallel.
- **True or False?**

This statement is **false** in \(\mathbb{R}^3\).

### Explanation:
In \(\mathbb{R}^2\) (two-dimensional space), the statement would be true because all lines parallel to a given line must be in the same plane, and if two lines are parallel to the same line, they must be parallel to each other.

However, in \(\mathbb{R}^3\), two lines can be parallel to a third line but **not parallel to each other**. This is because, in three-dimensional space, the two lines could lie in different planes. For example, imagine two lines that are both vertical and parallel to the z-axis, but one is on the x-axis and the other is on the y-axis. These two lines are parallel to the z-axis but are not parallel to each other, since they lie in different planes and do not have the same direction.

Thus, the correct answer is **False**.

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### Related questions:
1. What are the conditions for two lines to be parallel in \(\mathbb{R}^3\)?
2. Can two skew lines in \(\mathbb{R}^3\) be parallel to the same line?
3. How do you determine whether two lines are skew in \(\mathbb{R}^3\)?
4. What is the difference between parallel and skew lines in \(\mathbb{R}^3\)?
5. How can two lines be parallel in \(\mathbb{R}^2\) but not in \(\mathbb{R}^3\)?

### Tip:
In \(\mathbb{R}^3\), when analyzing line relationships, always check whether the lines lie in the same plane or different planes before concluding if they are parallel or skew.

Determine whether each statement is true or false in double-struck R3.
(a) Two lines parallel to a third line are parallel. True or False?

Explore whether two lines parallel to a third line are always parallel in 3D space (R^3). This question covers key geometric concepts and theorems in 3D coordinate space, providing insight into parallelism and planes. Suitable for students in Grades 10-12.

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