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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 \(\overline{BC}\), 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 \(\angle AYX\)?

ant101  Oct 8, 2018

Best Answer 

 #1
avatar+575 
+1

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 \(\Gamma \)\(\Rightarrow \) 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.

fiora  Oct 8, 2018
edited by fiora  Oct 8, 2018
 #1
avatar+575 
+1
Best Answer

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 \(\Gamma \)\(\Rightarrow \) 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.

fiora  Oct 8, 2018
edited by fiora  Oct 8, 2018
 #2
avatar+92461 
+2

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)

 

 

cool cool cool

CPhill  Oct 8, 2018

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