Perpendicular Lines

Dependency: Protractor axiom. A derived definition: use the measurable right angle to distinguish perpendicular lines.

Definition

Let lines $a$ and $b$ intersect at $O$, forming four angles.

If one of these four angles is a right angle, measured by Protractor axiom as $90^\circ$, then $a$ and $b$ are perpendicular, denoted by $a\perp b$. Each line is a perpendicular to the other, and $O$ is the foot of the perpendicular.

This definition does not rely on a drawing looking like a cross. It reduces perpendicularity to an exact measurable quantity: $90^\circ$.

Remarks

• One right angle is enough. The four angles are vertical or supplementary pairs, so if one is $90^\circ$, all four are $90^\circ$ by Linear pair sums to 180° and equality of vertical angles.
• Constructibility and uniqueness are theorems. Saying “the perpendicular through a point” assumes such a line exists and is unique. That fact is provided by Perpendicular from a point to a line exists and is unique, not by this definition.
• Perpendicularity is symmetric but not reflexive. $a\perp b$ if and only if $b\perp a$, while no line is perpendicular to itself.