1. Set up an x-y coordinate system: Consider a Cartesian coordinate system where each line can be represented by an equation in the form $y = mx + c$ or $x = k$.

2. Identify perpendicular lines: Two lines are perpendicular if the product of their slopes is $-1$. For example, if one line has a slope $m$, the perpendicular line will have a slope $-\frac{1}{m}$.

3. Consider lines parallel to the x- and y-axes: Lines parallel to the x-axis have the form $y = c$ (horizontal lines), and lines parallel to the y-axis have the form $x = k$ (vertical lines). These lines are always perpendicular to each other.

4. Pairing horizontal and vertical lines: For any given point $(a, a)$, the horizontal line through this point is $y = a$ and the vertical line through this point is $x = a$. These two lines are perpendicular and can be paired together.

5. Generalization: The argument can be extended to any point $(a, b)$ in the plane. The horizontal line through this point is $y = b$ and the vertical line through this point is $x = a$. These lines are also perpendicular and can be paired together.

6. Conclusion: Since every line parallel to the x-axis can be paired with a line parallel to the y-axis through some point $(a, b)$, it is possible to pair all such lines into perpendicular pairs.

However, this argument only covers lines parallel to the x- and y-axes. For lines with arbitrary slopes, the problem becomes more complex. Consider lines with slopes $m_1$ and $m_2$ such that $m_1 \cdot m_2 = -1$. There is no guarantee that every line can be paired with another line to satisfy this condition, as the number of lines with a specific slope is not necessarily equal to the number of lines with the negative reciprocal slope.

Therefore, it is not possible to pair all straight lines in a plane into pairs of perpendicular lines such that every line belongs to a single pair.

The final answer is False.