Expressing this angle in terms of other angles I want to express $\theta$ in terms of $\alpha$ and $\beta$ (or their tangents or other trig functions), but have no idea how to do so. Can someone please clarify? I just don't see the needed relations.

 A: 
That is a general and very know result called External Angle Theorem:
$$\angle A+\angle B+\angle C=180°\to\alpha+\beta+\angle C=180° \quad (1)$$
and also
$$\angle C+\theta=180° \to \angle C=180°-\theta\quad (2)$$
By $(1)$ and $(2)$ we get
$$\theta=\alpha +\beta$$
A: Let $D$ be the other endpoint of the line marked $b$, and let $\gamma = \text{m} \angle ADB$. Obviously $\theta + \gamma = \pi$, because $\theta$ and $\gamma$ add up to one side of the a straight line (the infinite line $AD$, namely). What do you know about $\alpha + \beta + \gamma$?
A: Let E be the end point of line marked as b.
And $\angle BEC = y$
Then in $\triangle BEC$
$y + \alpha + 90 = 180$
$y + \alpha = 90$
$y = 90 - \alpha$
Let $\angle AEC = z$
Then in $\triangle AEC$
$z + 90 + \beta = 180$
$z + \beta = 90$
$z = 90 - \beta$
Also,
$\theta + y + z = 180$
Putting the values of $y$ and $z$ from above equations,
$\theta + 90 - \alpha + 90 - \beta = 180$
$\theta = \alpha + \beta$
A: When a side of triangle is produced, external angle $ \theta $ so formed equals  sum of internal angles $ \alpha , \beta. $
