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In the figure, $PA$ is tangent to semicircle $SAR$, $PB$ is tangent to semicircle $RBT$, and $SRT$ is a straight line. If arc $AS$ is $58^\circ$ and arc $BT$ is $37^\circ$, then find $\angle APB$, in degrees.

 

 

 

Guest Jun 29, 2018
 #1
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Draw AR

Since arc AS  is 58°....then angle  ARS  also  = 58°

 

Similarly...draw  RB

And since arc BT is  37°, then angle BRT  also  = 37°

 

So...we have quadrilateral  PARB

 

Since   angle SRT  = 180°  ....then angle ARB  =

 

180  - angle ARS  - angle BRT  =  

180  - 58  - 37  =  85°

 

And  angles  RAP  and RBP  = 90°

 

And the sum of the interior angles of a quadrilateral  = 360°

 

So

angle ARB + angle RAP  + angle RBP  + angle APB  = 360

85  + 90  + 90 + angle APB  = 360

265  + APB  = 360

APB  = 95°

 

 

cool cool cool

CPhill  Jun 29, 2018

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