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Circle Gamma is the incircle of triangle ABC and is also the circumcircle of triangle XYZ. The point X is on segment BC, the point Y is on segment AB, and the point Z is on segment AC. If angle A=40 degrees,angle B=60 degrees, and angle C=80 degrees, what is the measure of angle YZX?

 Sep 12, 2018
 #1
avatar+100587 
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See the following  image :

 

 

AZDY  forms a  kite with angles AZD, AYD  = 90    

And  ZAY  = 40

So...angle ZDY =    360 - 2(90)  - 40  = 140

And in triangle ZDY, because DZ  = DY....then triangle ZDY is isosceles with angle YZD  = [ 180 - 140 ] / 2  = 

40/2   = 20

 

And CXDZ  also forms a kite with angles DXC, DZC  =  90

And angle ZCX  =  80

So....angle ZDX  =  360  - 2(90) - 80  =  100

And in triangle ZDX, because DX  = DZ....then triangle ZDX is isosceles with angle DZX  = [ 180 - 100 ] / 2  =

80/2   =  40

 

And angle YZX   = angle YZD  + angle DZX  =  20 + 40    =  60

 

 

cool cool cool 

 Sep 13, 2018

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