To completely determine a quadrilateral, you have to have five independent pieces of information, of sides and angles.
If you have five data (all outer sides and a diagonal), finding the angles is easy with the cosine rule.
Similarly with four sides and an angle, the cosine rule is enough to solve for the other angles.
With three sides and two angles, simple application of the cosine rule is enough to find the other sides and angles.
Except in one case. If the two angles are not in between any of the three sides, then there is no immediate way to use the cosine rule. How would the final side be solved for in this case?
Example (apologies for bad quality):
Best Answer
Rotate the figure such that the side with two angles lies on the $x$-axis, with length $k$. Finding the coordinates of the points, the opposite side must have length $5$, or:
$$(2 \sin 87 - 4 \sin 85)^2 + (k - 2 \cos 87 - 4 \cos 85)^2 = 25 $$ $$\implies k^2 +2(-2 \cos 87-4 \cos85)k + \left((-2 \cos87-4\cos85)^2 + (2 \sin 87 - 4 \sin 85)^2 - 25\right) = 0$$ $$\implies k \approx 5.041$$
The constant term can be simplified further as $4 + 16 + 8 \cos 87 \cos 85 - 8 \sin 87 \sin 85 - 25$ $ = 16 \cos 172 - 5$. This gives:
$$k = \frac{4(\cos 87 + 2 \cos 85) + \sqrt{16(\cos 87 + 2 \cos 85)^2 - 4(16 \cos 172 - 5)}}{2}$$
$$= 2(\cos 87 + 2 \cos 85) + \sqrt{4(\cos 87 + 2 \cos 85)^2 - 16 \cos 172 + 5}$$