Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular/Proof 1

Corollary to Two Angles on Straight Line make Two Right Angles

If a straight line meets another straight line, the bisectors of the two adjacent angles between them are perpendicular.

Proof

Figure

Let $AB$ and $CD$ be two straight lines that cross at $E$.

Let $\angle AEC$ be bisected by $EF$.

Let $\angle CEB$ be bisected by $EG$.

Thus:

$2 \angle FEC = \angle AEC$

and:

$2 \angle CEG = \angle CEB$

But from Two Angles on Straight Line make Two Right Angles, $\angle AEC + \angle CEB$ equal $2$ right angles.

Thus $2 \angle FEC + 2 \angle CEG$ equal $2$ right angles.

Hence $\angle FEG = \angle FEC + \angle CEG$ equals $1$ right angle.

That is $EF$ and $EG$ are perpendicular.

$\blacksquare$