+884 Circle \(\Gamma\) is the incircle of triangle ABC and is also the circumcircle of triangle XYZ. The point X is on \(\overline{BC}\), point Y is on \(\overline{AB}\), and the point Z is on \(\overline{AC}\). If \(\angle A=40^\circ, \angle B=60^\circ\)and \(\angle C=80^\circ\),what is the measure of \(\angle AYX\)?
+583 Assume Circle \(\Gamma \) is center at point \(\Gamma \). Connect \(\Gamma \) to point X and Y
Angle \(\Gamma \)XB = Angle \(\Gamma \)YB =90 degrees.
Angle X\(\Gamma \)Y= 360 - 60 - 90 - 90 =120 degree
And segment \(\Gamma \)X = segment \(\Gamma \)Y \(\Rightarrow \) angle \(\Gamma \)XY = angel \(\Gamma \)YX =1/2 *(180 degree - angel X\(\Gamma \)Y)=1/2(180-120）=30 degrees.
Angel AYX= angel AY\(\Gamma \) +angle \(\Gamma \)YX=90 degrees + 30 degrees= 120 degrees.
edited: to understand my answer, you need know the properties of inscrided circle(incircle) and angles. I urge you to draw picture to better understand my answer. You should check my answer ,because it might not be correct.
+583 Assume Circle \(\Gamma \) is center at point \(\Gamma \). Connect \(\Gamma \) to point X and Y
Angle \(\Gamma \)XB = Angle \(\Gamma \)YB =90 degrees.
Angle X\(\Gamma \)Y= 360 - 60 - 90 - 90 =120 degree
And segment \(\Gamma \)X = segment \(\Gamma \)Y \(\Rightarrow \) angle \(\Gamma \)XY = angel \(\Gamma \)YX =1/2 *(180 degree - angel X\(\Gamma \)Y)=1/2(180-120）=30 degrees.
Angel AYX= angel AY\(\Gamma \) +angle \(\Gamma \)YX=90 degrees + 30 degrees= 120 degrees.
edited: to understand my answer, you need know the properties of inscrided circle(incircle) and angles. I urge you to draw picture to better understand my answer. You should check my answer ,because it might not be correct.
+128408 Fiona's answer is correct
See the diagram :
Since in triangle XBY...angle XBY = 60
And since BX and BY are tangents to the incircle drawn from B....then BX = BY
And the angles opposite these sides are also equal = [ 180 - angle XBY ] / 2 =
[ 180 - 60 ] / 2 = 120 / 2 = 60
So angle BYX = angle BXY = 60
But angle BYX is supplemental to angle AYX = 180 - 60 = 120 (degrees)
