If the above diagrams are right, then the key is to draw an additional chord from the end if the chord to the other end of the first diagonal, making another right triangle that is similar to the one in question.
Now with solutiony goodness below
Triangle BDE is similar to ABC, therefore,
BE/BD = BC/AB, with BC=3/4*BD
Rearranging that, you end up with the ratio of AB to BC is 2/sqrt(6). The angle is ~35 degrees per the posts below
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