Find the relationship of the angles in the square below

Question

If ABCD is a square and AM=ME. Calculate $\dfrac{\alpha}{\theta}$ ($S=\dfrac{3}{2}$)

I tried to do the distribution of the angles but I can't find the relationship

Answer 1

Proceeding from your angle chasing, note that $\triangle AYD$ and $\triangle CDY$ are congruent. So, $\angle XAY=45^{\circ}-\angle YAD =45^{\circ}-\theta .$ As a result, $XYEA$ is a cyclic quadrilateral, which implies:

$$\angle JXY=\angle YEA \implies 45^{\circ}+\theta-\alpha =45^{\circ}-\frac{\theta}{2}\implies \frac{\alpha}{\theta}=\frac{3}{2}.$$