Adjust Angle to Add Vector for Non-Equiangular Non-Equilateral

Question

I asked this question: Adjust Angle to Add Vector and the solution showed that for equiangular, equilateral triangles the ratio between $\theta$ and $\phi$ was $\pi + \theta = 2\phi$:

But now I want to know, what if my angles aren't equiangular and my sides aren't equilateral can I still find a ratio between $\theta$ and $\phi + \psi$ when I'm adding one more side to circuit from $\vec{z}$'s head to tail?

Answer 1

We cannot create a simple ratio just between $\phi + \psi$ and $\theta$. A complex ratio can be obtained using the Law of Cosines and the lengths of the vectors.

Here's why:

These polygons share only two things:

1. A vertex at the intersection of $\vec{a}$ and $\vec{z}$ and $\vec{x}$ and $\vec{z}$
2. A vertex at the intersection of $\vec{c}$ and $\vec{z}$ and $\vec{y}$ and $\vec{z}$

Since a triangle's internal angles cannot be obtained from the length of a single side of a triangle is not enough information to establish a ratio between $\phi + \psi$ and $\theta$.

The original ratio could be derived because of the shared angles, and from two angles of a triangle you can always infer the third angle.

In the example below the orange triangles are identical, and all though the blue triangle shares two vertexes it is clear that a ratio cannot be established between the angles of the two triangles without more being shared between them: