In the figure, ABCD and BEFG are squares, and BCE is an equilateral triangle. What is the number of degrees in angle GCE ?

Draw a circle with B  as a center and BA as a radius....

This circle will pass through the points  A, C, E  and G

And angle EBG  will  be a central angle that will intercept the minor arc  EG

And the measure of this angle = 90°

And angle  GCE  will be an inscribed angle  intercepting the same arc as  angle EBG 

And an inscribed angle intercepting the same arc as a central angle will have 1/2 the measure of that angle......so.....  angle GCE  =  (1/2) (90°)  =  45°

Wow, that's really good!

I would not have thought to draw a circle ! 

Here is a picture, I thought I might as well share it:

Thanks for that pic, hectictar........your illustrations bring "life" to these answers....!!!

Not to beat a dead horse, but.....here's a way easier solution based on hectictar's pic.....no circle drawing is actually required!!!!

Note that triangle CBG is isosceles with angle  CBG  = 150

Since CB  = BG.....so...  angle BCG  =  [ 180 - 150] / 2   =  15

But angle BCE  =  60  =   angle BCG + angle GCE

So

60  = 15  + angle GCE  

45  =  angle GCE 

In the figure, ABCD and BEFG are squares, and BCE  is an equilateral triangle.

What is the number of degrees in angle GCE ?

Center of a circle in B with radius r=BG=BE=BC

Central Angle Theorem:

The angle at the centre is twice the angle at the circumference.

\(\beta = \frac{\alpha}{2}=\frac{90^{\circ}}{2}=45^{\circ}\)