An equilateral triangle ABC shares a common side BC with a square BCDE, as pictured. What is the number of degrees in angle DAE (not picture

For convenience sake...let the side of the square  = 1

Then the  height of the equilateral triangle  = √3/2

Position D  at  (0,0)

Then A will have the coordinates  ( 1/2, 1 + √3/2)

And we can find  1/2 of the measure of   angle  DAE  as

arctan [  (1/2) / ( 1 +√3/2 ) ]  =  15°

So...DAE  will be twice  this  = 30°

Actually....this could be done more simply

Draw  AD....now....we have ΔACD  with  angle ACD  = 150°

And since CD  = CA, then angle CAD  and And ADC  are equal

So angle CAD = [180 -150 ] / 2  = 30/2  =  15°

And if we draw, AE, by the same process,  in Δ ABE, angle EAB = 15°

So CAB  = 60°  =  angle CAD + angle DAE + angle EAB

 60   =  15  + angle DAE + 15

60  = 30 + angle DAE

30° =  angle DAE