I have two lines $AB$ and $BC$ with lengths $x$ and $y$, respectively.
$AB$ has a bearing $\theta$, $BC$ has a bearing $\phi$.
I need an equation that will provide the 'bearing' of $BC$ with respect to $AB$. Essentially, taking $AB$ as 'North' and recalculating the bearing of $BC$.
It's been a long time since I've done any trig and after a few hours of googling and trial and error I haven't gotten very far.
Assuming you're talking about a plane, I believe what you want is:
$$\text{RelativeBearing}(\theta, \phi) = \begin{cases} \phi - \theta, & \phi \geq \theta \\ 360^\circ - \theta + \phi, & \text{otherwise} \end{cases}$$
This gives a positive result if $\phi \geq \theta$ and a "looped-around" result otherwise.