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

tertre
Apr 18, 2017

#1**+2 **

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°

CPhill
Apr 18, 2017

#2**+2 **

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:

hectictar
Apr 18, 2017

#3**+1 **

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

CPhill
Apr 18, 2017

#4**+2 **

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

CPhill
Apr 18, 2017

#6**+1 **

**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.

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

heureka
Apr 18, 2017