How to find the equation of a line that makes an angle θ with another line with a known equation 
A better explanation of my problem

So I'm having a bit of a mathematical dilemma but I'm hoping someone can help me.
I want to know the equation of a line $g(x)$, and it shares an intersection $(9,3)$ with line $y=-0.25x + 5.25$. The intersection makes an angle of $38^\circ$. How might I work out the equation for $g(x)?$ Is there even enough to work with to make this possible?
You can assume that both lines are straight.

I haven't the slightest idea.
So I make it up from scratch. 
Lines have slopes. Slopes are like angles.  How do angles compare to slopes?
A slope is the "rise over the run" i.e. change in distance of the y value over the change in the x value.  That has to be proportional to $\sin/\cos$ or $\tan $ of the angle.
So $angle = \arctan slope $.
$Slope = \tan angle $.
That's just common sense.  So the angle of $y = -.25x +5.25$ is $\arctan -1/4 = -14$ degrees.
The angle of intersection is $38$  however the question if the intersection is $38$ more than $-14$ or less.  
I'll do both.  If the line of intersection is $38$ degrees more, The line I want has angle of $38 -14 =24$ so the line has a slope of $\tan 24 =.45$.
The line has point $(9,3) $ and slope $.45$ so the equation of the line is $y -3 = .45 (x -9) $ or $y =.45 x -1.05$
If the angle of intersection is $38$ degrees less, I want an angle of $-14 -38=-52$ or a slope of $\tan _52 = -1.28$
So the equation is $y-3=-1.28 (x-9)$ or $y =-1.28x +14.52$
