Find the angle between the diagonals in a quadrilateral

Question

Find the missing angle, $\theta$, between the diagonals in a quadrilateral. The angle $\angle ABC$ is right. I can find any other angle, except the angles between the diagonals. Any hint of how to find $\theta$?

Answer 1

The key observation is that $BD$ bisects $\measuredangle{ADC}$(I used a different naming of the vertices).

The hint to proving this: observe that if we draw perpendicular lines $BH$,$BI$,$BJ$ to $AD$,$AC$,$CD$, then $BA$ bisects $\measuredangle{HAI}$ and $BC$ bisects $\measuredangle{ICJ}.$

Answer 2

There are many quadrilaterals with all integer angles. For example this one. I doubt if there is always a "clever and simple" reasoning about the diagonal angle. Expressing the coordinates of C en D in terms of the angles and using the inproduct results -with use of a calculator- in the angles asked.