This problem is from Geometry Snacks by Ed Southall and Vincent Pantaloni – it’s a great book.
Two squares are constructed such that three vertices are collinear as shown. Find the value of the marked angle.
I started by marking in the right angles. And I added the diagonal of the larger square (pink line).
Because there are right angles at 
and 
, we know there is a circle, which has the diagonal of the square as its diameter (see second image below).
is 
(Angle between the diagonal and side of a square)
is a cyclic quadrilateral.
In cyclic quadrilaterals opposite angles are supplementary.
Hence, 
As 
, 
must be