3 reflections leads to glide In a previous problem I asked about the notation used here..  I'm still not sure how to show it even though I now get what it's asking.
The following is an exercise from The Four Pillars of Geometry.

3.6.5. Show that the reflections in lines $L$, $M$, and $N$ (in that order) have the same outcome as reflections in lines $L'$, $M'$, and $N$, where $M'$ is perpendicular to $N$

I've had to ask around to clairify parts of the problem; it should be noted that the lines $L'$ and $M'$ are essentially arbitrary with no relation to $L$ and $M$ except that they appear in the same order when you do the reflections.  Also, no picture is given as the lines are general though I've drawn a few pictures to help reason it out.
My main issue might be that I'm not sure what tools to use, so please be as specific as possible in your answer.
 A: I understand the question as in Mark's comment: $L$, $M$ and $N$ are given lines, and we want to show that there are lines $L'$ and $M'$ with $M'\perp N$ such that reflections in $L'$, $M'$ and $N$ (in that order) have the same effect as reflections in $L$, $M$ and $N$ (in that order).
First, since the reflection in $N$ is last in both cases, we can cancel it, so the task is to show that there are lines $L'$ and $M'$ such that reflections in $L'$ and $M'$ have the same effect as reflections in $L$ and $M$, subject to $M'\perp N$.
This is false. If $L$ and $M$ are parallel, reflections in $L$ and $M$ amount to a translation perpendicular to them by twice their distance, and this can only be reproduced by $L'$ and $M'$ if they, too, are parallel to $L$ and $M$; hence we cannot choose $M'$ perpendicular to $N$ unless $M$ already happens to be.
It's true, however, if $L$ and $M$ aren't parallel. In this case, it follows almost immediately from the hint given in the book right above the exercise:

Reflections in any two lines meeting at the same angle $\theta/2$ at the same point $P$ give the same outcome.

