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?

Let's break down the given information and deduce relationships between the lines:

Given conditions:

1. Line X is perpendicular to line Y (X | Y).
2. Line X is parallel to line Z (X || Z).
3. Line U is perpendicular to line V (U | V).
4. Line U is perpendicular to line W (U | W).
5. Line X is perpendicular to line V (X | V).

Deductions:

From (1) X | Y and (2) X || Z:

• If X is perpendicular to Y, and Z is parallel to X, then Z must also be perpendicular to Y (Z | Y).

From (3) U | V and (4) U | W:

• If two lines (V and W) are both perpendicular to a third line (U) in the same plane, then the two lines (V and W) must be parallel to each other (V || W).

From (1) X | Y and (5) X | V:

• If two lines (Y and V) are both perpendicular to a third line (X) in the same plane, then the two lines (Y and V) must be parallel to each other (Y || V).

Now let's evaluate the options:

A) Z, U and W are parallel We know U | W. This means U and W are not parallel. So, statement A is incorrect.

B) X, V and Y are parallel We know X | Y. This means X and Y are not parallel. So, statement B is incorrect.

C) Z, V and U are all perpendicular to W We deduced V || W. If V and W are parallel, they cannot be perpendicular (unless they are the same line, which is not implied). So, V is not perpendicular to W. Thus, statement C is incorrect.

D) Y, V and W are parallel From our deductions: * We found Y || V. * We found V || W. Combining these, if Y is parallel to V, and V is parallel to W, then Y, V, and W must all be parallel to each other (Y || V || W). This statement is correct.

The final answer is D