alpha=42.5degree
beta=35.2degree
a=18.0
I'm assuming side A is directly opposite angle $\displaystyle \alpha$, there is a side B opposite angle $\displaystyle \beta$, and there is a hypotenuse opposite angle $\displaystyle \gamma$.
If so...
The sum of the interior angles in a triangle equals 180, so:
$\displaystyle
\alpha + \beta + \gamma = 180
$
$\displaystyle
42.5 + 35.2 + \gamma = 180
$
$\displaystyle
\gamma = 102.3
$
The Law of Sines:
$\displaystyle
\frac{18}{\sin (42.5)} = \frac{B}{\sin (35.2)}
$
$\displaystyle
B = \frac{18\sin (35.2)}{\sin (42.5)}
$
$\displaystyle
\frac{18}{\sin (42.5)} = \frac{C}{\sin (102.3)}
$
$\displaystyle
C = \frac{18\sin (102.3)}{\sin (42.5)}
$
Check out the wikipedia entry on the law of sines:
Law of sines - Wikipedia, the free encyclopedia
What this problem is asking is known as triangulation:
Triangulation - Wikipedia, the free encyclopedia