Geometry Reflection Notation

Question

The following are exercises from The Four Pillars of Geometry; I'm not sure what they are stating, for example I don't know what the addition of prime (an apostrophe) to a line means. There are no examples or references.

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 $N$.

3.6.6 Next show that reflections in lines $L'$, $M'$, and $N$ have the same outcome as reflections in lines $L'$, $M''$, and $N'$ where $M''$ is parallel to $L'$ and $N'$ is perpendicular to $M''$.

3.6.7 Deduce from 3.6.6 that the combination of any three reflections is a glide reflection

I don't need to know how to do the problem, just what it's asking.

Answer 1

$L'$, $M'$, $M''$, and $N'$ are just more lines, with no specific known relationship to the original $L$, $M$, and $N$, except that they happen to take their places in the three-reflection composition.

Essentially, these are three steps to showing that the composite of any three reflections is a glide reflection. Let me rephrase the steps:

1. Given three arbitrary lines $\ell_1=L$, $\ell_2=M$, and $\ell_3=N$, let $T=R_{\ell_3}\circ R_{\ell_2}\circ R_{\ell_1}$ (reflect over $\ell_1$, then over $\ell_2$, then over $\ell_3$). Show that there exist some lines $\ell_4=L'$ and $\ell_5=M'$ with $\ell_3\perp\ell_5$ such that $R_{\ell_3}\circ R_{\ell_5}\circ R_{\ell_4}=T$.

2. Given the above, show that there exist some lines $\ell_6=M''$ and $\ell_7=N'$ with $\ell_6\parallel\ell_4$ and $\ell_7\perp\ell_6$ such that $R_{\ell_7}\circ R_{\ell_6}\circ R_{\ell_4}=T$ (probably through $R_{\ell_3}\circ R_{\ell_5}\circ R_{\ell_4}=T$ from part 1).

3. Conclude that the composite of any three reflections (our original $T$) is a glide reflection—that is, we know from the previous two parts that $T=R_{\ell_7}\circ R_{\ell_6}\circ R_{\ell_4}$, so explain why $R_{\ell_7}\circ R_{\ell_6}\circ R_{\ell_4}$ a glide reflection.

(Please let me know if this is still confusing. I am now sure that there is not a missing diagram or some other context that you have left out, but the original notation is definitely hard to follow and I'm not sure that my notation is any better.)