In a plane, line X is perpendicular to line Y and parallel to line Z; line U is perpendicular to both lines V and W; line X is perpendicular to line V.

Which one of the following statements is correct?

The correct option is "Y, V and W are parallel."

Let's analyze the relationship between the lines step-by-step to understand why this statement is correct. All lines lie in the same two-dimensional plane.

Step 1: Understand the relationships of lines X, Y, and Z

We are given that line X is perpendicular to line Y ($X\perp Y$) and parallel to line Z ($X\parallel Z$).
If two lines are parallel, any line perpendicular to one must also be perpendicular to the other.
Since X is perpendicular to Y, and Z is parallel to X, it follows that line Z must also be perpendicular to line Y:
$Z\perp Y$

Step 2: Understand the relationships of lines U, V, and W

We are given that line U is perpendicular to both lines V and W ($U\perp V$ and $U\perp W$).
In a plane, if two lines are perpendicular to the same line, they must be parallel to each other.
Since both V and W are perpendicular to U, we can conclude that line V and line W are parallel:
$V\parallel W$

Step 3: Connect the two sets of lines using the relationship between X and V

We are given that line X is perpendicular to line V ($X\perp V$).
Now we can relate the directions of all the lines:
1. We know that $X\perp Y$ (given) and $X\perp V$ (given).
In a plane, if two lines (Y and V) are perpendicular to the same line (X), they must be parallel to each other:
$Y\parallel V$

2. We already established from Step 2 that V is parallel to W:
$V\parallel W$

3. By the transitive property of parallel lines in a plane, since Y is parallel to V, and V is parallel to W, then Y, V, and W must all be parallel to one another:
$Y\parallel V\parallel W$

Therefore, the statement "Y, V and W are parallel" is correct.