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

Consider a line l on which Y and Z lies. Now, a point X away from l such that XY⊥l and Z is a point on line l other than Y. In ΔXYZ, ∠Y = 90°. So, in ΔXYZ, →∠YXZ + ∠XZY + ∠XYZ = 180°. Putting ∠XYZ = 90° →∠YXZ + ∠XZY = 90° →∠X + ∠Z = 90° →∠Z < 90° →∠Z < ∠Y →XY < XZ (Side opposite to greater angle is greater) XY is the shortest of all line segments from X to l.