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?
+128460 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
