Solving a set of Geometry Constraints - What's that Method called?
Its called a geometric constraints solver (a good primer on subject). You can find a open source solver as part of Open Cascade but its a bit convoluted to get going. A simpler solution for just solving, but also 3D solver capable, is geosolver.

Making your own (algebraic solution being easiest to write) is also not that hard, just a bit of work to make it automatically write the equations (and be efficient at solving). Take the following drawing as a example (made and solved with Mathematica):



You can simply conclude that its a sum of vectors that has to loop back at the origin thus:

$$
7\begin{pmatrix}\sin(15^\circ)\\ \cos(15^\circ)\end{pmatrix}+5\begin{pmatrix}\sin(120^\circ)\\ \cos(120^\circ)\end{pmatrix}+2\begin{pmatrix}\sin(195^\circ)\\ \cos(195^\circ)\end{pmatrix}+4\begin{pmatrix}\sin(\delta)\\ \cos(\delta)\end{pmatrix}+x\begin{pmatrix}\sin(-90^\circ)\\ \cos(-90^\circ)\end{pmatrix}=0
$$

And solve the 2 unknowns from 2 equations, either algebraically or by using a gradient descent solver of some kind.