Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular/Proof 1
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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
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.
$\blacksquare$