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

From ProofWiki
Jump to navigation Jump to search

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.

$\blacksquare$


Sources