Example.
Let $l$ and $m$ be two different lines intersecting at a point $P$. Show that $R_m R_l =R_{p, \theta}$ where $B$ is twice the directed angle $\alpha$ from $l$ to $m$.
Solution.
Let $h$ be the line through $P$ such that the directed angle from $h$ to $l$ is $\alpha$, and let $n$ be the line through P such that the directed angle from $m$ to $n$ is $\alpha $, as in the figure above.
Let $A$ be a point on $h$ and let $C$ be a point on $l$. Note that $A, C$, and $P$ cannot be collinear. If they were collinear, then $h = l$ , and it would follow that $l$, $h$, and $m$ are all the same line, which is a contradiction.
Consider the effect of $R_m R_l $ on the points $A, P$, and $C$. It is clear that $R_{P, \theta}$ has exactly the same effect, so by Theorem 8.2.8, $R_m R_l = R_{P, \theta}$ ∎
There is something obscure concerning why $l = m$ coincide. If $l=m$ coincide, how can this be proven?