Unique Perpendicular Theorem

Theorem: A perpendicular drawn to a line at a point is unique.

Prerequisites:
Angle on a straight line (proof)

Proof:

Let AB be a straight line. If possible, let OM and ON be the two distinct perpendiculars drawn to AB at the point O. Hence, the angles, $\angle MOA$ and $\angle NOB$ are $90^o$ each.

Since the total angle on a straight line AB at any point is $180^o$,

$\therefore\;\;\angle MOA + \angle MON + \angle NOB = 180^o$
$\therefore\;\;90^o + \angle MON + 90^o = 180^o$
$\therefore\;\;\angle MON = 0^o$

But this is a contradiction, as OM and ON are distinct.

Hence, distinct perpendiculars are not possible, and only unique perpendicular can be drawn at a point on a line.


Recommended:
Unique parallel theorem
Angle sum property of a triangle
Exterior angle property