Below $\measuredangle$ means measure of an angle mod $180^{\circ}$. When measuring angles ,clockwise measurement is taken negative and wise versa.
If $\measuredangle ABC=\measuredangle DEF$ doesn't imply that $\angle ABC=\angle DEF$
I have problem in understanding following statement.
One thing we have to be careful about is that $2\measuredangle ABC = 2\measuredangle XYZ$ does not imply $\measuredangle ABC =\measuredangle XYZ$, because we are taking angles modulo $180^{\circ}$. Hence it does not make sense to take half of a directed angle.
Are the lines from EGMO by Evan Chen.
I don't quite understand this. Is there any example to show that this statement is correct ?
An example is enough. Thanks in advance.
Post-script: There is no correct tag for this question. Will anyone help in finding one helpful tag.
Here’s a simple example: take $\measuredangle{ABC}=135°$ and $\measuredangle{XYZ}=45°$. Then $$2\measuredangle{ABC}=270°-180°=90°=2\measuredangle{XYZ}.$$ More generally, if $\measuredangle{ABC}$ and $\measuredangle{XYZ}$ differ by 90°, then doubling them will produce the same angle mod $180°$.