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Lines $XQ$ and $XR$ are tangent to a circle, as shown below.   If $\angle QXR = 60^\circ$, then find $\angle TAU$.

 

 Dec 29, 2023
 #1
avatar+126665 
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Let O be the center of the  circle

Angles  OTX and OUX = 90°

Angle TXU =  angle QXR = 60°

 

So   TOUX is   a quadrilateral  whose interior angles  sum to  360°

 

So angle   TOU  =   360 - 90 - 90 -60  = 120°  = minor arc  TU

 

And angle TAU is an inscribed  angle intercepting the  same  minor arc TU

 

So TAU measures  (1/2) (120)  = 60°

 

cool cool cool

 Dec 29, 2023

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