Show that of all line segments drawn from a give point not on it, the perpendicular line segment is the shortest.

Given: x is a line and A is a point not lying on x.

C is any point on x other than B.

To prove: AB < AC
Proof:
In \(\triangle{ABC}\), \(\angle{B}\) is the right angle.

Therefore, \(\angle{C}\) is an acute angle.
\(\Rightarrow \) \(\angle{C}\) < \(\angle{B}\)
\(\therefore \) AB < AC
(Since, side opposite to greater angle is longer)

Hence, proved that the perpendicular line segment is the shortest.