Disorganized Pinwheel

Start labeling one of the unknown angles $x$ and expressing the rest of the unknown angles in terms of $x$ , as done in the figure above. Apply the law of sines to all the small triangles to get:

$\frac{a}{sin(30)}= \frac{b}{sin(x)}$

$\frac{a}{sin(130-x)}= \frac{d}{sin(20)}$

$\frac{c}{sin(110-x)}= \frac{b}{sin(40)}$

$\frac{c}{sin(50)}= \frac{d}{sin(x-20)}$

Solve for $\frac{b}{d}$ to get:

$\frac{b}{d}=\frac{sin(130-x)}{sin(20)}\times\frac{sin(x)}{sin(30)}= \frac{sin(50)}{sin(x-20)}\times\frac{sin(40)}{sin(110-x)}$

Equation on the right has five solutions, but only two of those are positive. They are $x=30^\circ$ and $x=100^\circ$ . They correspond to acute angles between the diagonals of $60^\circ$ and $50^\circ$ .